8.6 Gamma functions.nb

(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 7.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[    379972,       7830]
NotebookOptionsPosition[    367480,       7508]
NotebookOutlinePosition[    369055,       7560]
CellTagsIndexPosition[    369012,       7557]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[Cell[TextData[StyleBox["Rules for integrands involving gamma \
functions",
  FontFamily->"Arial"]], "None"]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.4965234353062396`*^9, {3.5193214825316973`*^9, 3.5193214852617016`*^9}, {
   3.523316399894187*^9, 3.5233164032341914`*^9}, {3.523318350558202*^9, 
   3.5233183516382036`*^9}, {3.536542725780803*^9, 3.536542725780803*^9}, {
   3.5367764878952584`*^9, 3.5367764886452594`*^9}, {3.544322815209103*^9, 
   3.5443228192963104`*^9}, {3.5450662624913826`*^9, 3.545066292141424*^9}, {
   3.545066402271579*^9, 3.5450664054415827`*^9}, {3.5450834884295015`*^9, 
   3.5450834895895033`*^9}, {3.5454961263762493`*^9, 
   3.5454961321462574`*^9}, {3.5454963863166127`*^9, 
   3.5454964580867133`*^9}, {3.545497046127537*^9, 3.545497046127537*^9}, {
   3.5454971130676303`*^9, 3.545497136257663*^9}, 3.545610399977621*^9, {
   3.546040778592962*^9, 3.546040778592962*^9}, {3.5461052065706367`*^9, 
   3.5461052249474688`*^9}, {3.546191339879622*^9, 3.5461913430596266`*^9}, {
   3.5462141824067917`*^9, 3.5462142145768366`*^9}, {3.5462145455473003`*^9, 
   3.5462145601073203`*^9}, {3.5463149785974817`*^9, 
   3.5463149794195285`*^9}, {3.5488738437683954`*^9, 
   3.5488738699084315`*^9}, {3.5488890647645535`*^9, 3.548889080804576*^9}, {
   3.5505951583099127`*^9, 3.5505951620351257`*^9}, {3.5758457803910656`*^9, 
   3.5758457938510847`*^9}, {3.5758458780912023`*^9, 
   3.5758458797312045`*^9}, {3.5758460250714083`*^9, 3.575846028691413*^9}, {
   3.5758461031815176`*^9, 3.575846103681518*^9}, {3.5758463018517957`*^9, 
   3.5758463042917986`*^9}, {3.5758464155019546`*^9, 3.575846417191957*^9}, {
   3.5758468610025787`*^9, 3.5758468696025906`*^9}, {3.5758469815427475`*^9, 
   3.5758469940427647`*^9}, {3.575847092102902*^9, 3.575847093372904*^9}, {
   3.5758472070230627`*^9, 3.5758472111630683`*^9}},
 TextAlignment->Center,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["1.",
   FontFamily->"Arial"]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{"u", " ", 
    RowBox[{"Gamma", "[", 
     RowBox[{"n", ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9}, 
   3.574964614077545*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599651210365`*^9, 
   3.57495996777104*^9}, {3.574961504633192*^9, 3.574961504973192*^9}, 
   3.57496462482756*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by parts", "Subsubsection",
 CellChangeTimes->{3.4941724567488375`*^9}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    SubscriptBox["\[PartialD]", "x"], 
    RowBox[{"Gamma", "[", 
     RowBox[{"n", ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]}], "]"}]}], "\[Equal]", 
   RowBox[{"-", 
    FractionBox[
     RowBox[{"b", " ", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "x"}]}], ")"}], 
       RowBox[{"n", "-", "1"}]]}], 
     SuperscriptBox["\[ExponentialE]", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]]]}]}]],
  CellChangeTimes->{{3.721077959964218*^9, 3.72107796559554*^9}, {
   3.7210781170462027`*^9, 3.721078164012889*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.7210781295989203`*^9, 3.72107813133502*^9}, {
   3.72107817365144*^9, 3.7210781742474747`*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494439179913333*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    RowBox[{"Gamma", "[", 
     RowBox[{"n", ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", "x"}]}], ")"}], " ", 
        RowBox[{"Gamma", "[", 
         RowBox[{"n", ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", "x"}]}]}], "]"}]}], "b"]}]}]}]}], "+", 
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}], ")"}], "n"], 
     SuperscriptBox["\[ExponentialE]", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]]], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", " ", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", "x"}]}], ")"}], " ", 
        RowBox[{"Gamma", "[", 
         RowBox[{"n", ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", "x"}]}]}], "]"}]}], "b"]}]}]}]}], "-", 
  FractionBox[
   RowBox[{"Gamma", "[", 
    RowBox[{
     RowBox[{"n", "+", "1"}], ",", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "x"}]}]}], "]"}], "b"]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944391698033185`*^9, 3.4944392275633993`*^9, {
   3.7210780475842295`*^9, 3.7210780582708406`*^9}, {3.7210781941136103`*^9, 
   3.7210782613824577`*^9}, 3.721078305663991*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{"Gamma", "[", 
     RowBox[{"n_", ",", 
      RowBox[{"a_.", "+", 
       RowBox[{"b_.", "*", "x_"}]}]}], "]"}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{"a", "+", 
       RowBox[{"b", "*", "x"}]}], ")"}], "*", 
     RowBox[{
      RowBox[{"Gamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", "x"}]}]}], "]"}], "/", "b"}]}], " ", "-", " ", 
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{
       RowBox[{"n", "+", "1"}], ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", "*", "x"}]}]}], "]"}], "/", "b"}]}], " ", "/;", "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "b", ",", "n"}], "}"}], ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.4944391618433075`*^9, {
   3.721096929542216*^9, 3.7210969343364897`*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["2.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"d", " ", "x"}], ")"}], "m"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"b", " ", "x"}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9}, 
   3.574964614077545*^9, {3.574964922777977*^9, 3.574964926547982*^9}, 
   3.574971287956888*^9, {3.574971361966992*^9, 3.574971373617008*^9}, 
   3.709308508646453*^9, {3.7210769791321173`*^9, 3.7210769801441755`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["1.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9}, 
   3.574964614077545*^9, {3.574964922777977*^9, 3.574964926547982*^9}, 
   3.574971287956888*^9, {3.574971361966992*^9, 3.574971373617008*^9}, 
   3.7092687343291826`*^9, 3.709308499522931*^9, 3.7210769828293295`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["1.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"n", "\[Element]", "\[DoubleStruckCapitalZ]"}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9}, 
   3.574964614077545*^9, {3.574964922777977*^9, 3.574964926547982*^9}, 
   3.574971287956888*^9, {3.574971361966992*^9, 3.574971373617008*^9}, 
   3.7092687343291826`*^9, 3.709308499522931*^9, {3.7093096690149946`*^9, 
   3.7093096776574097`*^9}, 3.7210769838043847`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{"0", ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749647914277935`*^9, 
   3.5749647966178007`*^9}, {3.5749649281379848`*^9, 3.574964954528022*^9}, {
   3.5749713766370125`*^9, 3.574971413477064*^9}, 3.57498495665808*^9, {
   3.5749852211284504`*^9, 3.57498530673857*^9}, {3.709268738465401*^9, 
   3.7092687851164837`*^9}, 3.7093097077518625`*^9, 3.721076984466423*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"Gamma", "[", 
    RowBox[{"0", ",", "z"}], "]"}], "\[Equal]", 
   RowBox[{"ExpIntegralE", "[", 
    RowBox[{"1", ",", "z"}], "]"}]}]],
  CellDingbat->None,
  CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
    3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
    3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
    3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
    3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
    3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
    3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
    3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
    3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
    3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
    3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
    3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9}, 
    3.4944395044737873`*^9, {3.5525056981388483`*^9, 
    3.5525057072180643`*^9}, {3.552505897366798*^9, 3.552505924198845*^9}, {
    3.5525060501222663`*^9, 3.5525060654882936`*^9}, {3.5525061289648046`*^9, 
    3.5525061418972273`*^9}, {3.5525062448730087`*^9, 
    3.5525062459650106`*^9}, {3.5525100606611214`*^9, 
    3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
    3.57497143191709*^9, 3.5749714489771137`*^9}, 3.5749714856671653`*^9, 
    3.709268822873951*^9, 3.7092690350265255`*^9, {3.7092691334905024`*^9, 
    3.7092691676751623`*^9}},
  TextAlignment->Center,
  FontSize->12,
  FontWeight->"Bold"]
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9, 
   3.5525065695719786`*^9}, {3.5749853271085987`*^9, 3.5749853271085987`*^9}, 
   3.7092691223832827`*^9, {3.7210767172471385`*^9, 3.721076734526127*^9}, {
   3.721076771490241*^9, 3.721076771490241*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9, 
   3.5525065695719786`*^9}, {3.5749853271085987`*^9, 3.5749853271085987`*^9}, 
   3.7092691223832827`*^9}],

Cell[BoxData[{
 RowBox[{"\[Integral]", 
  RowBox[{
   FractionBox[
    RowBox[{"Gamma", "[", 
     RowBox[{"0", ",", 
      RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     RowBox[{"\[Integral]", 
      RowBox[{
       FractionBox[
        RowBox[{"ExpIntegralE", "[", 
         RowBox[{"1", ",", 
          RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
       RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}], "\[IndentingNewLine]", 
 RowBox[{"  ", 
  RowBox[{
   RowBox[{
    RowBox[{"\[LongRightArrow]", "  ", "b"}], " ", "x", " ", 
    RowBox[{"HypergeometricPFQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"1", ",", "1", ",", "1"}], "}"}], ",", 
      RowBox[{"{", 
       RowBox[{"2", ",", "2", ",", "2"}], "}"}], ",", 
      RowBox[{
       RowBox[{"-", "b"}], " ", "x"}]}], "]"}]}], "-", 
   RowBox[{"EulerGamma", " ", 
    RowBox[{"Log", "[", "x", "]"}]}], "-", 
   RowBox[{
    FractionBox["1", "2"], " ", 
    SuperscriptBox[
     RowBox[{"Log", "[", 
      RowBox[{"b", " ", "x"}], "]"}], "2"]}]}]}]}], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9}, 
   3.4944395044737873`*^9, {3.5525056981388483`*^9, 3.5525057072180643`*^9}, {
   3.552505897366798*^9, 3.552505924198845*^9}, {3.5525060501222663`*^9, 
   3.5525060654882936`*^9}, {3.5525061289648046`*^9, 
   3.5525061418972273`*^9}, {3.5525062448730087`*^9, 
   3.5525062459650106`*^9}, {3.5525100606611214`*^9, 
   3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
   3.57497143191709*^9, 3.5749714489771137`*^9}, 3.5749714856671653`*^9, 
   3.709268822873951*^9, 3.7092690350265255`*^9, {3.7092691334905024`*^9, 
   3.7092691676751623`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"0", ",", 
       RowBox[{"b_.", "*", "x_"}]}], "]"}], "/", "x_"}], ",", "x_Symbol"}], 
   "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"b", "*", "x", "*", 
     RowBox[{"HypergeometricPFQ", "[", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"1", ",", "1", ",", "1"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"2", ",", "2", ",", "2"}], "}"}], ",", 
       RowBox[{
        RowBox[{"-", "b"}], "*", "x"}]}], "]"}]}], " ", "-", " ", 
    RowBox[{"EulerGamma", "*", 
     RowBox[{"Log", "[", "x", "]"}]}], " ", "-", " ", 
    RowBox[{
     RowBox[{"1", "/", "2"}], "*", 
     RowBox[{
      RowBox[{"Log", "[", 
       RowBox[{"b", "*", "x"}], "]"}], "^", "2"}]}]}], " ", "/;", "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{"b", ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494439298133498*^9, {
   3.5525057121476727`*^9, 3.5525057235356927`*^9}, 3.5525065386059246`*^9, {
   3.5525100827663603`*^9, 3.5525101007999916`*^9}, {3.574971543937247*^9, 
   3.574971565597277*^9}, {3.57498488505798*^9, 3.5749849154980226`*^9}, {
   3.5749849658780932`*^9, 3.5749849692580976`*^9}, 3.7092692026232243`*^9, 
   3.7092692984867935`*^9, {3.709269335120658*^9, 3.7092693386774645`*^9}, 
   3.7093099799381447`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["x:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{"1", ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749647914277935`*^9, 
   3.5749647966178007`*^9}, {3.5749649281379848`*^9, 3.574964954528022*^9}, {
   3.5749713766370125`*^9, 3.574971413477064*^9}, 3.57498495665808*^9, {
   3.5749852211284504`*^9, 3.57498530673857*^9}, {3.709268738465401*^9, 
   3.709268751309224*^9}, {3.709269573021481*^9, 3.7092696283899784`*^9}, {
   3.70930932164629*^9, 3.70930932164629*^9}, {3.709309641442289*^9, 
   3.709309642060324*^9}, {3.709309711308669*^9, 3.7093097369933147`*^9}, {
   3.709309836585691*^9, 3.709309863294938*^9}, {3.709314505135638*^9, 
   3.7093145123896513`*^9}, {3.709315130402023*^9, 3.709315130402023*^9}, {
   3.709315482388446*^9, 3.7093154865108533`*^9}, {3.709315831200208*^9, 
   3.7093158320122547`*^9}, 3.72107698494745*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["\<\
Derivation: Algebraic expansion\
\>", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.709313827569145*^9, 3.7093138311883516`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"Gamma", "[", 
    RowBox[{"1", ",", "z"}], "]"}], "\[Equal]", 
   FractionBox["1", 
    SuperscriptBox["\[ExponentialE]", "z"]]}]], "Input",
  CellChangeTimes->{{3.70931399439364*^9, 3.709314069157575*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.709313827569145*^9, 3.709313849221983*^9}, {
   3.709313956843973*^9, 3.7093139746904044`*^9}, 3.709314105050639*^9, {
   3.709315648373751*^9, 3.709315648373751*^9}, {3.7210785613906174`*^9, 
   3.7210785632457237`*^9}}],

Cell[TextData[{
 "Note: ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " automatically evaluates ",
 Cell[BoxData[
  RowBox[{"Gamma", "[", 
   RowBox[{"1", ",", "z"}], "]"}]],
  CellChangeTimes->{{3.70931399439364*^9, 3.709314069157575*^9}}],
 " to ",
 Cell[BoxData[
  SuperscriptBox["\[ExponentialE]", 
   RowBox[{"-", "z"}]]],
  CellChangeTimes->{{3.70931399439364*^9, 3.709314069157575*^9}}],
 "."
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.709313827569145*^9, 3.709313849221983*^9}, {
   3.709313956843973*^9, 3.7093139746904044`*^9}, 3.709314105050639*^9, {
   3.709315648373751*^9, 3.709315648373751*^9}, {3.7093156864559293`*^9, 
   3.709315761858242*^9}, {3.721078568914048*^9, 3.721078593625461*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"n", ">", "1"}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9, 
   3.5525065695719786`*^9}, {3.5749853271085987`*^9, 
   3.5749853271085987`*^9}, {3.7092696407919316`*^9, 
   3.7092696407919316`*^9}, {3.7093096482556787`*^9, 
   3.7093096482556787`*^9}, {3.7093097441793275`*^9, 
   3.7093097441793275`*^9}, {3.709309827146674*^9, 3.709309827146674*^9}, {
   3.7093145217506676`*^9, 3.7093145217506676`*^9}, {3.7093151539554653`*^9, 
   3.7093151539554653`*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   FractionBox[
    RowBox[{"Gamma", "[", 
     RowBox[{"1", ",", 
      RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     RowBox[{"\[Integral]", 
      RowBox[{
       FractionBox["1", 
        RowBox[{"x", " ", 
         SuperscriptBox["\[ExponentialE]", 
          RowBox[{"b", " ", "x"}]]}]], 
       RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9}, 
   3.4944395044737873`*^9, {3.5525056981388483`*^9, 3.5525057072180643`*^9}, {
   3.552505897366798*^9, 3.552505924198845*^9}, {3.5525060501222663`*^9, 
   3.5525060654882936`*^9}, {3.5525061289648046`*^9, 
   3.5525061418972273`*^9}, {3.5525062448730087`*^9, 
   3.5525062459650106`*^9}, {3.5525100606611214`*^9, 
   3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
   3.57497143191709*^9, 3.5749714489771137`*^9}, 3.5749714856671653`*^9, 
   3.709309376380128*^9, 3.709309407253894*^9, {3.7093094939968557`*^9, 
   3.7093095366912975`*^9}, {3.709314182883684*^9, 3.7093142012777367`*^9}, {
   3.709314257078928*^9, 3.7093143190864744`*^9}, {3.709315528139693*^9, 
   3.7093155364857073`*^9}, 3.709315658726343*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
   RowBox[{"Int", "[", 
    RowBox[{
     RowBox[{
      RowBox[{"Gamma", "[", 
       RowBox[{"1", ",", 
        RowBox[{"b_.", "*", "x_"}]}], "]"}], "/", "x_"}], ",", "x_Symbol"}], 
    "]"}], " ", ":=", "\n", "  ", 
   RowBox[{
    RowBox[{"Int", "[", 
     RowBox[{
      RowBox[{"1", "/", 
       RowBox[{"(", 
        RowBox[{"x", "*", 
         RowBox[{"E", "^", 
          RowBox[{"(", 
           RowBox[{"b", "*", "x"}], ")"}]}]}], ")"}]}], ",", "x"}], "]"}], 
    " ", "/;", "\n", 
    RowBox[{"FreeQ", "[", 
     RowBox[{"b", ",", "x"}], "]"}]}]}], " ", "*)"}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494439298133498*^9, {
   3.5525057121476727`*^9, 3.5525057235356927`*^9}, 3.5525065386059246`*^9, {
   3.5525100827663603`*^9, 3.5525101007999916`*^9}, {3.574971543937247*^9, 
   3.574971565597277*^9}, {3.57498488505798*^9, 3.5749849154980226`*^9}, {
   3.5749849658780932`*^9, 3.5749849692580976`*^9}, {3.709269510121369*^9, 
   3.709269531263406*^9}, {3.709269655223957*^9, 3.709269660215966*^9}, {
   3.7093097950756173`*^9, 3.709309814697852*^9}, 3.7093098960599957`*^9, 
   3.7093099903861847`*^9, {3.709314677233594*^9, 3.709314693239222*^9}, 
   3.709315156139469*^9, {3.70931578105534*^9, 3.7093158262069225`*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{"n", "-", "1"}], "\[Element]", 
       SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]]], "None"], Cell[
     "", "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749647914277935`*^9, 
   3.5749647966178007`*^9}, {3.5749649281379848`*^9, 3.574964954528022*^9}, {
   3.5749713766370125`*^9, 3.574971413477064*^9}, 3.57498495665808*^9, {
   3.5749852211284504`*^9, 3.57498530673857*^9}, {3.709268738465401*^9, 
   3.709268751309224*^9}, {3.709269573021481*^9, 3.7092696283899784`*^9}, {
   3.70930932164629*^9, 3.70930932164629*^9}, {3.709309641442289*^9, 
   3.709309642060324*^9}, {3.709309711308669*^9, 3.7093097369933147`*^9}, {
   3.709309836585691*^9, 3.709309863294938*^9}, {3.709314505135638*^9, 
   3.7093145123896513`*^9}, {3.709315130402023*^9, 3.709315130402023*^9}, {
   3.7093158693593907`*^9, 3.709315877689867*^9}, 3.721076985430478*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["\<\
Derivation: Algebraic expansion\
\>", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.709313827569145*^9, 3.7093138311883516`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"Gamma", "[", 
    RowBox[{"n", ",", "z"}], "]"}], "\[Equal]", 
   RowBox[{
    FractionBox[
     SuperscriptBox["z", 
      RowBox[{"n", "-", "1"}]], 
     SuperscriptBox["\[ExponentialE]", "z"]], "+", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{"n", "-", "1"}], ")"}], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{
       RowBox[{"n", "-", "1"}], ",", "z"}], "]"}]}]}]}]], "Input",
  CellChangeTimes->{{3.70931399439364*^9, 3.709314069157575*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.709313827569145*^9, 3.709313849221983*^9}, {
   3.709313956843973*^9, 3.7093139746904044`*^9}, 3.709314105050639*^9, {
   3.7210785467547803`*^9, 3.7210785467547803`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"n", "-", "1"}], "\[Element]", 
   SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9, 
   3.5525065695719786`*^9}, {3.5749853271085987`*^9, 
   3.5749853271085987`*^9}, {3.7092696407919316`*^9, 
   3.7092696407919316`*^9}, {3.7093096482556787`*^9, 
   3.7093096482556787`*^9}, {3.7093097441793275`*^9, 
   3.7093097441793275`*^9}, {3.709309827146674*^9, 3.709309827146674*^9}, {
   3.7093145217506676`*^9, 3.7093145217506676`*^9}, {3.7093151539554653`*^9, 
   3.7093151539554653`*^9}, {3.709315883525201*^9, 3.709315883525201*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", "b"}]}], " ", 
    RowBox[{"\[Integral]", 
     RowBox[{
      FractionBox[
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"b", " ", "x"}], ")"}], 
        RowBox[{"n", "-", "2"}]], 
       SuperscriptBox["\[ExponentialE]", 
        RowBox[{"b", " ", "x"}]]], 
      RowBox[{"\[DifferentialD]", "x"}]}]}]}]}], "+", 
  RowBox[{
   RowBox[{"(", 
    RowBox[{"n", "-", "1"}], ")"}], " ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{
        RowBox[{"n", "-", "1"}], ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", 
      RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
       RowBox[{"-", 
        RowBox[{"Gamma", "[", 
         RowBox[{
          RowBox[{"n", "-", "1"}], ",", 
          RowBox[{"b", " ", "x"}]}], "]"}]}]}]}]}]}]}], "+", 
  RowBox[{
   RowBox[{"(", 
    RowBox[{"n", "-", "1"}], ")"}], " ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{
        RowBox[{"n", "-", "1"}], ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9}, 
   3.4944395044737873`*^9, {3.5525056981388483`*^9, 3.5525057072180643`*^9}, {
   3.552505897366798*^9, 3.552505924198845*^9}, {3.5525060501222663`*^9, 
   3.5525060654882936`*^9}, {3.5525061289648046`*^9, 
   3.5525061418972273`*^9}, {3.5525062448730087`*^9, 
   3.5525062459650106`*^9}, {3.5525100606611214`*^9, 
   3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
   3.57497143191709*^9, 3.5749714489771137`*^9}, 3.5749714856671653`*^9, 
   3.709309376380128*^9, 3.709309407253894*^9, {3.7093094939968557`*^9, 
   3.7093095366912975`*^9}, {3.709314182883684*^9, 3.7093142012777367`*^9}, {
   3.709314257078928*^9, 3.7093143190864744`*^9}, {3.721079203283332*^9, 
   3.7210792256896133`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"b_.", "*", "x_"}]}], "]"}], "/", "x_"}], ",", "x_Symbol"}], 
   "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"-", 
     RowBox[{"Gamma", "[", 
      RowBox[{
       RowBox[{"n", "-", "1"}], ",", 
       RowBox[{"b", "*", "x"}]}], "]"}]}], " ", "+", " ", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{"n", "-", "1"}], ")"}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"Gamma", "[", 
         RowBox[{
          RowBox[{"n", "-", "1"}], ",", 
          RowBox[{"b", "*", "x"}]}], "]"}], "/", "x"}], ",", "x"}], "]"}]}]}],
    " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{"b", ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"IGtQ", "[", 
     RowBox[{"n", ",", "1"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494439298133498*^9, {
   3.5525057121476727`*^9, 3.5525057235356927`*^9}, 3.5525065386059246`*^9, {
   3.5525100827663603`*^9, 3.5525101007999916`*^9}, {3.574971543937247*^9, 
   3.574971565597277*^9}, {3.57498488505798*^9, 3.5749849154980226`*^9}, {
   3.5749849658780932`*^9, 3.5749849692580976`*^9}, {3.709269510121369*^9, 
   3.709269531263406*^9}, {3.709269655223957*^9, 3.709269660215966*^9}, {
   3.7093097950756173`*^9, 3.709309814697852*^9}, 3.7093098960599957`*^9, 
   3.7093099903861847`*^9, {3.709314677233594*^9, 3.709314693239222*^9}, 
   3.709315156139469*^9, 3.7093158618059587`*^9, 3.7210793674497213`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["3:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"n", "\[Element]", 
       SuperscriptBox["\[DoubleStruckCapitalZ]", "-"]}]]]], "None"], Cell[
     "", "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749647914277935`*^9, 
   3.5749647966178007`*^9}, {3.5749649281379848`*^9, 3.574964954528022*^9}, {
   3.5749713766370125`*^9, 3.574971413477064*^9}, 3.57498495665808*^9, {
   3.5749852211284504`*^9, 3.57498530673857*^9}, {3.709268738465401*^9, 
   3.709268751309224*^9}, {3.709269573021481*^9, 3.7092696283899784`*^9}, {
   3.70930932164629*^9, 3.70930932164629*^9}, {3.709309641442289*^9, 
   3.709309642060324*^9}, {3.709309711308669*^9, 3.7093097369933147`*^9}, {
   3.709309836585691*^9, 3.709309863294938*^9}, {3.709314505135638*^9, 
   3.7093145123896513`*^9}, {3.7093149657341213`*^9, 
   3.7093149657341213`*^9}, {3.7093152082927613`*^9, 
   3.7093152092911634`*^9}, {3.70931589941611*^9, 3.709315905395452*^9}, 
   3.7210769859085054`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["\<\
Derivation: Algebraic expansion\
\>", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.709313827569145*^9, 3.7093138311883516`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"Gamma", "[", 
    RowBox[{"n", ",", "z"}], "]"}], "\[Equal]", 
   RowBox[{
    RowBox[{"-", 
     FractionBox[
      SuperscriptBox["z", "n"], 
      RowBox[{"n", " ", 
       SuperscriptBox["\[ExponentialE]", "z"]}]]}], "+", 
    RowBox[{
     FractionBox["1", "n"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{
       RowBox[{"n", "+", "1"}], ",", "z"}], "]"}]}]}]}]], "Input",
  CellChangeTimes->{{3.70931399439364*^9, 3.709314069157575*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.709313827569145*^9, 3.709313849221983*^9}, {
   3.709313956843973*^9, 3.7093139746904044`*^9}, 3.709314105050639*^9, {
   3.709314936658269*^9, 3.709314936658269*^9}, {3.7210779123294935`*^9, 
   3.7210779123294935`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"n", "\[Element]", 
   SuperscriptBox["\[DoubleStruckCapitalZ]", "-"]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9, 
   3.5525065695719786`*^9}, {3.5749853271085987`*^9, 
   3.5749853271085987`*^9}, {3.7092696407919316`*^9, 
   3.7092696407919316`*^9}, {3.7093096482556787`*^9, 
   3.7093096482556787`*^9}, {3.7093097441793275`*^9, 
   3.7093097441793275`*^9}, {3.709309827146674*^9, 3.709309827146674*^9}, {
   3.7093145217506676`*^9, 3.7093145217506676`*^9}, {3.709314992942569*^9, 
   3.709314992942569*^9}, {3.709315917875166*^9, 3.709315917875166*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      RowBox[{"-", 
       FractionBox["b", "n"]}]}]}], " ", 
    RowBox[{"\[Integral]", 
     RowBox[{
      FractionBox[
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"b", " ", "x"}], ")"}], 
        RowBox[{"n", "-", "1"}]], 
       SuperscriptBox["\[ExponentialE]", 
        RowBox[{"b", " ", "x"}]]], 
      RowBox[{"\[DifferentialD]", "x"}]}]}]}]}], "+", 
  RowBox[{
   FractionBox["1", "n"], " ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{
        RowBox[{"n", "+", "1"}], ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", 
      RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
       FractionBox[
        RowBox[{"Gamma", "[", 
         RowBox[{"n", ",", 
          RowBox[{"b", " ", "x"}]}], "]"}], "n"]}]}]}]}]}], "+", 
  RowBox[{
   FractionBox["1", "n"], " ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{
        RowBox[{"n", "+", "1"}], ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9}, 
   3.4944395044737873`*^9, {3.5525056981388483`*^9, 3.5525057072180643`*^9}, {
   3.552505897366798*^9, 3.552505924198845*^9}, {3.5525060501222663`*^9, 
   3.5525060654882936`*^9}, {3.5525061289648046`*^9, 
   3.5525061418972273`*^9}, {3.5525062448730087`*^9, 
   3.5525062459650106`*^9}, {3.5525100606611214`*^9, 
   3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
   3.57497143191709*^9, 3.5749714489771137`*^9}, 3.5749714856671653`*^9, 
   3.709309376380128*^9, 3.709309407253894*^9, {3.7093094939968557`*^9, 
   3.7093095366912975`*^9}, {3.709314182883684*^9, 3.7093142012777367`*^9}, {
   3.709314257078928*^9, 3.7093143190864744`*^9}, {3.7093147527739267`*^9, 
   3.7093148299400635`*^9}, {3.7210791338263593`*^9, 3.7210791346824083`*^9}, 
   3.7210794249600105`*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"b_.", "*", "x_"}]}], "]"}], "/", "x_"}], ",", "x_Symbol"}], 
   "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"b", "*", "x"}]}], "]"}], "/", "n"}], " ", "+", " ", 
    RowBox[{
     RowBox[{"1", "/", "n"}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"Gamma", "[", 
         RowBox[{
          RowBox[{"n", "+", "1"}], ",", 
          RowBox[{"b", "*", "x"}]}], "]"}], "/", "x"}], ",", "x"}], "]"}]}]}],
    " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{"b", ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"ILtQ", "[", 
     RowBox[{"n", ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494439298133498*^9, {
   3.5525057121476727`*^9, 3.5525057235356927`*^9}, 3.5525065386059246`*^9, {
   3.5525100827663603`*^9, 3.5525101007999916`*^9}, {3.574971543937247*^9, 
   3.574971565597277*^9}, {3.57498488505798*^9, 3.5749849154980226`*^9}, {
   3.5749849658780932`*^9, 3.5749849692580976`*^9}, {3.709269510121369*^9, 
   3.709269531263406*^9}, {3.709269655223957*^9, 3.709269660215966*^9}, {
   3.7093097950756173`*^9, 3.709309814697852*^9}, 3.7093098960599957`*^9, 
   3.7093099903861847`*^9, {3.709314677233594*^9, 3.709314693239222*^9}, {
   3.709315017575012*^9, 3.7093150184174137`*^9}, 3.709315066791499*^9, 
   3.709315920850336*^9, 3.721079472304719*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"n", "\[NotElement]", "\[DoubleStruckCapitalZ]"}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749647914277935`*^9, 
   3.5749647966178007`*^9}, {3.5749649281379848`*^9, 3.574964954528022*^9}, {
   3.5749713766370125`*^9, 3.574971413477064*^9}, 3.57498495665808*^9, {
   3.5749852211284504`*^9, 3.57498530673857*^9}, {3.709268738465401*^9, 
   3.709268751309224*^9}, {3.709269573021481*^9, 3.7092696283899784`*^9}, {
   3.70930992662545*^9, 3.7093099304630566`*^9}, 3.721076986433535*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"n", "\[NotElement]", "\[DoubleStruckCapitalZ]"}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9, 
   3.5525065695719786`*^9}, {3.5749853271085987`*^9, 
   3.5749853271085987`*^9}, {3.7092696407919316`*^9, 
   3.7092696407919316`*^9}, {3.7093099377794695`*^9, 3.7093099377794695`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"b", " ", "x"}]}], "]"}], "x"], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      RowBox[{"Gamma", "[", "n", "]"}]}]}], " ", 
    RowBox[{"Log", "[", "x", "]"}]}]}], "-", 
  RowBox[{
   FractionBox[
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"b", " ", "x"}], ")"}], "n"], 
    SuperscriptBox["n", "2"]], 
   RowBox[{"HypergeometricPFQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"n", ",", "n"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"1", "+", "n"}], ",", 
       RowBox[{"1", "+", "n"}]}], "}"}], ",", 
     RowBox[{
      RowBox[{"-", "b"}], " ", "x"}]}], "]"}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9}, 
   3.4944395044737873`*^9, {3.5525056981388483`*^9, 3.5525057072180643`*^9}, {
   3.552505897366798*^9, 3.552505924198845*^9}, {3.5525060501222663`*^9, 
   3.5525060654882936`*^9}, {3.5525061289648046`*^9, 
   3.5525061418972273`*^9}, {3.5525062448730087`*^9, 
   3.5525062459650106`*^9}, {3.5525100606611214`*^9, 
   3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
   3.57497143191709*^9, 3.5749714489771137`*^9}, 3.5749714856671653`*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"b_.", "*", "x_"}]}], "]"}], "/", "x_"}], ",", "x_Symbol"}], 
   "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"Gamma", "[", "n", "]"}], "*", 
     RowBox[{"Log", "[", "x", "]"}]}], " ", "-", " ", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"b", "*", "x"}], ")"}], "^", "n"}], "/", 
      RowBox[{"n", "^", "2"}]}], "*", 
     RowBox[{"HypergeometricPFQ", "[", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"n", ",", "n"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{
         RowBox[{"1", "+", "n"}], ",", 
         RowBox[{"1", "+", "n"}]}], "}"}], ",", 
       RowBox[{
        RowBox[{"-", "b"}], "*", "x"}]}], "]"}]}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"b", ",", "n"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"Not", "[", 
     RowBox[{"IntegerQ", "[", "n", "]"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494439298133498*^9, {
   3.5525057121476727`*^9, 3.5525057235356927`*^9}, 3.5525065386059246`*^9, {
   3.5525100827663603`*^9, 3.5525101007999916`*^9}, {3.574971543937247*^9, 
   3.574971565597277*^9}, {3.57498488505798*^9, 3.5749849154980226`*^9}, {
   3.5749849658780932`*^9, 3.5749849692580976`*^9}, {3.709269510121369*^9, 
   3.709269531263406*^9}, {3.709269655223957*^9, 3.709269660215966*^9}, {
   3.7093099426652784`*^9, 3.709309944989682*^9}, 3.7093100003566017`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"d", " ", "x"}], ")"}], "m"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"b", " ", "x"}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"m", "\[NotEqual]", 
       RowBox[{"-", "1"}]}]]]], "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749647914277935`*^9, 
   3.5749647966178007`*^9}, {3.5749649281379848`*^9, 3.574964954528022*^9}, {
   3.5749713766370125`*^9, 3.574971378337015*^9}, 3.7093084809338675`*^9, 
   3.7210769869455643`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["\<\
Derivation: Integration by parts and piecewise constant extraction\
\>", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.5487821419713373`*^9, 3.5487821496713486`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    SubscriptBox["\[PartialD]", "x"], 
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"d", " ", "x"}], ")"}], "m"], 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"b", " ", "x"}], ")"}], "m"]]}], "\[Equal]", "0"}]],
  CellChangeTimes->{{3.5487821700513773`*^9, 3.5487821882814026`*^9}}]
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494439945074404*^9, {
   3.5487821579913597`*^9, 3.548782159081362*^9}, {3.5487821973014154`*^9, 
   3.548782197851416*^9}, {3.709307809043476*^9, 3.709307809043476*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{"-", 
     FractionBox["1", "b"]}], 
    RowBox[{
     SubscriptBox["\[PartialD]", "x"], 
     RowBox[{"Gamma", "[", 
      RowBox[{
       RowBox[{"m", "+", "n", "+", "1"}], ",", 
       RowBox[{"b", " ", "x"}]}], "]"}]}]}], "\[Equal]", 
   FractionBox[
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"b", " ", "x"}], ")"}], 
     RowBox[{"m", "+", "n"}]], 
    SuperscriptBox["\[ExponentialE]", 
     RowBox[{"b", " ", "x"}]]]}]],
  CellChangeTimes->{{3.5487821700513773`*^9, 3.5487821882814026`*^9}}]
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494439945074404*^9, {
   3.5487821579913597`*^9, 3.548782159081362*^9}, {3.5487821973014154`*^9, 
   3.548782197851416*^9}, {3.709307809043476*^9, 3.709307809043476*^9}, {
   3.709308286552246*^9, 3.709308286552246*^9}}],

Cell["\<\
Note: The antiderivative is given directly without recursion so it is \
expressed entirely in terms of the incomplete gamma function without need for \
the exponential function.\
\>", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494439945074404*^9, {
   3.5525062989739037`*^9, 3.552506478514619*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"m", "\[NotEqual]", 
   RowBox[{"-", "1"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9, 
   3.5525065695719786`*^9}}],

Cell[BoxData[{
 RowBox[{
  RowBox[{
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"d", " ", "x"}], ")"}], "m"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"b", " ", "x"}]}], "]"}], 
     RowBox[{"\[DifferentialD]", 
      RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
       FractionBox[
        RowBox[{
         SuperscriptBox[
          RowBox[{"(", 
           RowBox[{"d", " ", "x"}], ")"}], 
          RowBox[{"m", "+", "1"}]], " ", 
         RowBox[{"Gamma", "[", 
          RowBox[{"n", ",", 
           RowBox[{"b", " ", "x"}]}], "]"}]}], 
        RowBox[{"d", " ", 
         RowBox[{"(", 
          RowBox[{"m", "+", "1"}], ")"}]}]]}]}]}]}], "+", 
   RowBox[{
    FractionBox["1", 
     RowBox[{"m", "+", "1"}]], " ", 
    RowBox[{"\[Integral]", 
     RowBox[{
      FractionBox[
       RowBox[{
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"d", " ", "x"}], ")"}], "m"], " ", 
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"b", " ", "x"}], ")"}], "n"]}], 
       SuperscriptBox["\[ExponentialE]", 
        RowBox[{"b", " ", "x"}]]], 
      RowBox[{"\[DifferentialD]", "x"}]}]}]}]}], 
  "                "}], "\[IndentingNewLine]", 
 RowBox[{"                                  ", 
  RowBox[{
   RowBox[{"\[LongRightArrow]", "  ", 
    FractionBox[
     RowBox[{
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"d", " ", "x"}], ")"}], 
       RowBox[{"m", "+", "1"}]], " ", 
      RowBox[{"Gamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"b", " ", "x"}]}], "]"}]}], 
     RowBox[{"d", " ", 
      RowBox[{"(", 
       RowBox[{"m", "+", "1"}], ")"}]}]]}], "+", 
   RowBox[{
    FractionBox[
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"d", " ", "x"}], ")"}], "m"], 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"m", "+", "1"}], ")"}], 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"b", " ", "x"}], ")"}], "m"]}]], " ", 
    RowBox[{"\[Integral]", 
     RowBox[{
      FractionBox[
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"b", " ", "x"}], ")"}], 
        RowBox[{"m", "+", "n"}]], 
       SuperscriptBox["\[ExponentialE]", 
        RowBox[{"b", " ", "x"}]]], 
      RowBox[{"\[DifferentialD]", 
       RowBox[{
       "x", "\[IndentingNewLine]", 
        "                                          ", "\[LongRightArrow]", 
        "  ", 
        FractionBox[
         RowBox[{
          SuperscriptBox[
           RowBox[{"(", 
            RowBox[{"d", " ", "x"}], ")"}], 
           RowBox[{"m", "+", "1"}]], " ", 
          RowBox[{"Gamma", "[", 
           RowBox[{"n", ",", 
            RowBox[{"b", " ", "x"}]}], "]"}]}], 
         RowBox[{"d", " ", 
          RowBox[{"(", 
           RowBox[{"m", "+", "1"}], ")"}]}]]}]}]}]}]}], "-", 
   FractionBox[
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"d", " ", "x"}], ")"}], "m"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{
       RowBox[{"m", "+", "n", "+", "1"}], ",", 
       RowBox[{"b", " ", "x"}]}], "]"}]}], 
    RowBox[{"b", " ", 
     RowBox[{"(", 
      RowBox[{"m", "+", "1"}], ")"}], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"b", " ", "x"}], ")"}], "m"]}]]}]}]}], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9}, 
   3.4944395044737873`*^9, {3.5525056981388483`*^9, 3.5525057072180643`*^9}, {
   3.552505897366798*^9, 3.552505924198845*^9}, {3.5525060501222663`*^9, 
   3.5525060654882936`*^9}, {3.5525061289648046`*^9, 
   3.5525061418972273`*^9}, {3.5525062448730087`*^9, 
   3.5525062459650106`*^9}, {3.5525100606611214`*^9, 
   3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
   3.7093073835012946`*^9, 3.709307411583344*^9}, 3.709307476558859*^9, {
   3.7093075575702024`*^9, 3.7093075697850237`*^9}, {3.709308069742737*^9, 
   3.709308098531788*^9}, {3.709308304617078*^9, 3.709308327689519*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"d_.", "*", "x_"}], ")"}], "^", "m_."}], "*", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"b_.", "*", "x_"}]}], "]"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"d", "*", "x"}], ")"}], "^", 
      RowBox[{"(", 
       RowBox[{"m", "+", "1"}], ")"}]}], "*", 
     RowBox[{
      RowBox[{"Gamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"b", "*", "x"}]}], "]"}], "/", 
      RowBox[{"(", 
       RowBox[{"d", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}]}], " ", "-", " ", "\n", 
    "  ", 
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"d", "*", "x"}], ")"}], "^", "m"}], "*", 
     RowBox[{
      RowBox[{"Gamma", "[", 
       RowBox[{
        RowBox[{"m", "+", "n", "+", "1"}], ",", 
        RowBox[{"b", "*", "x"}]}], "]"}], "/", 
      RowBox[{"(", 
       RowBox[{"b", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}], "*", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"b", "*", "x"}], ")"}], "^", "m"}]}], ")"}]}]}]}], " ", "/;",
    "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"b", ",", "d", ",", "m", ",", "n"}], "}"}], ",", "x"}], "]"}], 
    " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{"m", ",", 
      RowBox[{"-", "1"}]}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494439298133498*^9, {
   3.5525057121476727`*^9, 3.5525057235356927`*^9}, 3.5525065386059246`*^9, {
   3.5525100827663603`*^9, 3.5525101007999916`*^9}, 3.7061416388783164`*^9, {
   3.7093083795736103`*^9, 3.709308438215424*^9}, 3.709310009747819*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["3.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9}, 
   3.574964614077545*^9, {3.574964922777977*^9, 3.574964926547982*^9}, 
   3.574971287956888*^9, 3.709308513902753*^9, {3.7093101696196337`*^9, 
   3.7093101700636587`*^9}, {3.7210770302870436`*^9, 3.7210770305670595`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{"b", " ", "c"}], "-", 
        RowBox[{"a", " ", "d"}]}], "\[Equal]", "0"}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.574964815567827*^9, 3.5749648195978327`*^9}, 
   3.574965002088088*^9, {3.5749650378381386`*^9, 3.5749650380781384`*^9}, {
   3.5749850938482723`*^9, 3.574985134328329*^9}, {3.7093086668295*^9, 
   3.7093086793042135`*^9}, 3.7093101790071707`*^9, {3.721077069840306*^9, 
   3.7210770701493235`*^9}, {3.721077211412403*^9, 3.721077211412403*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["\<\
Derivation: Integration by substitution\
\>", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.721077214419575*^9, 3.721077217037725*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{"b", " ", "c"}], "-", 
    RowBox[{"a", " ", "d"}]}], "\[Equal]", "0"}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5749851409783382`*^9, 
   3.5749851409783382`*^9}, {3.709308692338058*^9, 3.709308692338058*^9}, {
   3.721077222861058*^9, 3.721077222861058*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
   RowBox[{"Gamma", "[", 
    RowBox[{"n", ",", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "x"}]}]}], "]"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox["1", "b"]}]}], " ", 
   RowBox[{"Subst", "[", 
    RowBox[{
     RowBox[{"\[Integral]", 
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         FractionBox[
          RowBox[{"d", " ", "x"}], "b"], ")"}], "m"], " ", 
       RowBox[{"Gamma", "[", 
        RowBox[{"n", ",", "x"}], "]"}], 
       RowBox[{"\[DifferentialD]", "x"}]}]}], ",", "x", ",", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "x"}]}]}], "]"}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944393519535737`*^9, {3.4944398753843064`*^9, 
   3.494439898614339*^9}, {3.5487729084881315`*^9, 3.548772951029406*^9}, {
   3.548773033491151*^9, 3.548773102989273*^9}, {3.548773240924715*^9, 
   3.5487732690827646`*^9}, {3.7093101834524245`*^9, 
   3.7093102025425167`*^9}, {3.7210772339756937`*^9, 3.721077280667364*^9}, 
   3.7210773303172045`*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"a_", "+", 
        RowBox[{"b_.", "*", "x_"}]}]}], "]"}]}], ",", "x_Symbol"}], "]"}], 
  " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"1", "/", "b"}], "*", 
    RowBox[{"Subst", "[", 
     RowBox[{
      RowBox[{"Int", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"(", 
           RowBox[{"d", "*", 
            RowBox[{"x", "/", "b"}]}], ")"}], "^", "m"}], "*", 
         RowBox[{"Gamma", "[", 
          RowBox[{"n", ",", "x"}], "]"}]}], ",", "x"}], "]"}], ",", "x", ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", "*", "x"}]}]}], "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "m", ",", "n"}], "}"}],
       ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"EqQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"b", "*", "c"}], "-", 
       RowBox[{"a", "*", "d"}]}], ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, {3.494439298133498*^9, 
   3.4944393224735327`*^9}, 3.548773313979644*^9, {3.548774218001231*^9, 
   3.548774225130444*^9}, {3.548779276211317*^9, 3.548779290343337*^9}, 
   3.5487837127235365`*^9, {3.5488095573285336`*^9, 3.5488095656285453`*^9}, {
   3.552514591145564*^9, 3.5525145954255695`*^9}, {3.552514725435752*^9, 
   3.5525147288657565`*^9}, {3.6144377669286184`*^9, 
   3.6144378180275407`*^9}, {3.6144389616939545`*^9, 3.614438967274274*^9}, 
   3.7061416389083166`*^9, {3.7093085949753904`*^9, 3.7093086239820495`*^9}, {
   3.7093102455339756`*^9, 3.7093103001330986`*^9}, 3.7093253569351254`*^9, 
   3.7210770537283845`*^9, {3.7210773637641172`*^9, 3.7210773677533455`*^9}, 
   3.7210774030243626`*^9, 3.7210774593365836`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "x"}]}]}], "]"}], 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}]], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{"n", "-", "1"}], "\[Element]", 
       SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]]], "None"], Cell[
     "", "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749647914277935`*^9, 
   3.5749647966178007`*^9}, {3.5749649281379848`*^9, 3.574964954528022*^9}, {
   3.5749713766370125`*^9, 3.574971413477064*^9}, 3.57498495665808*^9, {
   3.5749852211284504`*^9, 3.57498530673857*^9}, {3.709268738465401*^9, 
   3.709268751309224*^9}, {3.709269573021481*^9, 3.7092696283899784`*^9}, {
   3.70930932164629*^9, 3.70930932164629*^9}, {3.709309641442289*^9, 
   3.709309642060324*^9}, {3.709309711308669*^9, 3.7093097369933147`*^9}, {
   3.709309836585691*^9, 3.709309863294938*^9}, {3.709314505135638*^9, 
   3.7093145123896513`*^9}, {3.709315130402023*^9, 3.709315130402023*^9}, {
   3.7093158693593907`*^9, 3.709315877689867*^9}, 3.721076985430478*^9, {
   3.721079508541792*^9, 3.7210795345282784`*^9}, 3.721079711435396*^9, {
   3.7210799074776096`*^9, 3.721079907700622*^9}, {3.721079954882321*^9, 
   3.7210799551553364`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Algebraic expansion", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.709313827569145*^9, 3.7093138311883516`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"Gamma", "[", 
    RowBox[{"n", ",", "z"}], "]"}], "\[Equal]", 
   RowBox[{
    FractionBox[
     SuperscriptBox["z", 
      RowBox[{"n", "-", "1"}]], 
     SuperscriptBox["\[ExponentialE]", "z"]], "+", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{"n", "-", "1"}], ")"}], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{
       RowBox[{"n", "-", "1"}], ",", "z"}], "]"}]}]}]}]], "Input",
  CellChangeTimes->{{3.70931399439364*^9, 3.709314069157575*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.709313827569145*^9, 3.709313849221983*^9}, {
   3.709313956843973*^9, 3.7093139746904044`*^9}, 3.709314105050639*^9, {
   3.7210785467547803`*^9, 3.7210785467547803`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"n", "-", "1"}], "\[Element]", 
   SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5525065667951736`*^9, 
   3.5525065695719786`*^9}, {3.5749853271085987`*^9, 
   3.5749853271085987`*^9}, {3.7092696407919316`*^9, 
   3.7092696407919316`*^9}, {3.7093096482556787`*^9, 
   3.7093096482556787`*^9}, {3.7093097441793275`*^9, 
   3.7093097441793275`*^9}, {3.709309827146674*^9, 3.709309827146674*^9}, {
   3.7093145217506676`*^9, 3.7093145217506676`*^9}, {3.7093151539554653`*^9, 
   3.7093151539554653`*^9}, {3.709315883525201*^9, 3.709315883525201*^9}, {
   3.7210799190522714`*^9, 3.7210799193712893`*^9}, {3.7210799585855327`*^9, 
   3.721079958878549*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    FractionBox[
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}]], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      RowBox[{"\[Integral]", 
       RowBox[{
        FractionBox[
         SuperscriptBox[
          RowBox[{"(", 
           RowBox[{"a", "+", 
            RowBox[{"b", " ", "x"}]}], ")"}], 
          RowBox[{"n", "-", "1"}]], 
         RowBox[{
          RowBox[{"(", 
           RowBox[{"c", "+", 
            RowBox[{"d", " ", "x"}]}], ")"}], " ", 
          SuperscriptBox["\[ExponentialE]", 
           RowBox[{"a", "+", 
            RowBox[{"b", " ", "x"}]}]]}]], 
        RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}], "+", 
  RowBox[{
   RowBox[{"(", 
    RowBox[{"n", "-", "1"}], ")"}], " ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{
        RowBox[{"n", "-", "1"}], ",", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "x"}]}]}], "]"}], 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9}, 
   3.4944395044737873`*^9, {3.5525056981388483`*^9, 3.5525057072180643`*^9}, {
   3.552505897366798*^9, 3.552505924198845*^9}, {3.5525060501222663`*^9, 
   3.5525060654882936`*^9}, {3.5525061289648046`*^9, 
   3.5525061418972273`*^9}, {3.5525062448730087`*^9, 
   3.5525062459650106`*^9}, {3.5525100606611214`*^9, 
   3.5525100714875402`*^9}, {3.5749713237369385`*^9, 3.574971334336953*^9}, {
   3.57497143191709*^9, 3.5749714489771137`*^9}, 3.5749714856671653`*^9, 
   3.709309376380128*^9, 3.709309407253894*^9, {3.7093094939968557`*^9, 
   3.7093095366912975`*^9}, {3.709314182883684*^9, 3.7093142012777367`*^9}, {
   3.709314257078928*^9, 3.7093143190864744`*^9}, {3.721079203283332*^9, 
   3.7210792256896133`*^9}, {3.7210795385455074`*^9, 
   3.7210795544704185`*^9}, {3.7210796035102234`*^9, 3.721079620185177*^9}, {
   3.721079707635179*^9, 3.7210797250581756`*^9}, 3.721079761063235*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", "x_"}]}]}], "]"}], "/", 
     RowBox[{"(", 
      RowBox[{"c_.", "+", 
       RowBox[{"d_.", "*", "x_"}]}], ")"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"Int", "[", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", "*", "x"}]}], ")"}], "^", 
        RowBox[{"(", 
         RowBox[{"n", "-", "1"}], ")"}]}], "/", 
       RowBox[{"(", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "*", 
         RowBox[{"E", "^", 
          RowBox[{"(", 
           RowBox[{"a", "+", 
            RowBox[{"b", "*", "x"}]}], ")"}]}]}], ")"}]}], ",", "x"}], "]"}], 
    " ", "+", " ", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{"n", "-", "1"}], ")"}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"Gamma", "[", 
         RowBox[{
          RowBox[{"n", "-", "1"}], ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", "x"}]}]}], "]"}], "/", 
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", "*", "x"}]}], ")"}]}], ",", "x"}], "]"}]}]}], " ", "/;",
    "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"a", ",", "b", ",", "c", ",", "d"}], "}"}], ",", "x"}], "]"}], 
    " ", "&&", " ", 
    RowBox[{"IGtQ", "[", 
     RowBox[{"n", ",", "1"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494439298133498*^9, {
   3.5525057121476727`*^9, 3.5525057235356927`*^9}, 3.5525065386059246`*^9, {
   3.5525100827663603`*^9, 3.5525101007999916`*^9}, {3.574971543937247*^9, 
   3.574971565597277*^9}, {3.57498488505798*^9, 3.5749849154980226`*^9}, {
   3.5749849658780932`*^9, 3.5749849692580976`*^9}, {3.709269510121369*^9, 
   3.709269531263406*^9}, {3.709269655223957*^9, 3.709269660215966*^9}, {
   3.7093097950756173`*^9, 3.709309814697852*^9}, 3.7093098960599957`*^9, 
   3.7093099903861847`*^9, {3.709314677233594*^9, 3.709314693239222*^9}, 
   3.709315156139469*^9, 3.7093158618059587`*^9, 3.7210793674497213`*^9, {
   3.7210798446380153`*^9, 3.7210798812071066`*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["3:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"m", "\[Element]", 
          SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}], "\[Or]", " ", 
         RowBox[{"n", "\[Element]", 
          SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}], "\[Or]", " ", 
         RowBox[{
          RowBox[{"(", 
           RowBox[{"m", "|", "n"}], ")"}], "\[Element]", 
          "\[DoubleStruckCapitalZ]"}]}], ")"}], " ", "\[And]", " ", 
       RowBox[{"m", "\[NotEqual]", 
        RowBox[{"-", "1", "  "}]}]}]]]], "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.574964815567827*^9, 3.5749648195978327`*^9}, 
   3.574965002088088*^9, {3.5749650378381386`*^9, 3.5749650380781384`*^9}, {
   3.5749850938482723`*^9, 3.574985134328329*^9}, {3.7093086668295*^9, 
   3.7093086793042135`*^9}, 3.7093101790071707`*^9, {3.721077069840306*^9, 
   3.7210770701493235`*^9}, {3.72107742011034*^9, 3.721077420347354*^9}, {
   3.7210800014369836`*^9, 3.721080001757002*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by parts", "Subsubsection",
 CellChangeTimes->{3.4941724567488375`*^9}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    SubscriptBox["\[PartialD]", "x"], 
    RowBox[{"Gamma", "[", 
     RowBox[{"n", ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]}], "]"}]}], "\[Equal]", 
   RowBox[{"-", 
    FractionBox[
     RowBox[{"b", " ", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "x"}]}], ")"}], 
       RowBox[{"n", "-", "1"}]]}], 
     SuperscriptBox["\[ExponentialE]", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]]]}]}]],
  CellChangeTimes->{{3.7210786990344906`*^9, 3.721078737038664*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.7210787457181606`*^9, 3.7210787547866793`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{"(", 
    RowBox[{
     RowBox[{"m", "\[Element]", 
      SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}], "\[Or]", " ", 
     RowBox[{"n", "\[Element]", 
      SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}], "\[Or]", " ", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"m", "|", "n"}], ")"}], "\[Element]", 
      "\[DoubleStruckCapitalZ]"}]}], ")"}], " ", "\[And]", " ", 
   RowBox[{"m", "\[NotEqual]", 
    RowBox[{"-", "1", "  "}]}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5749851409783382`*^9, 
   3.5749851409783382`*^9}, {3.709308692338058*^9, 3.709308692338058*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    RowBox[{"Gamma", "[", 
     RowBox[{"n", ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], 
         RowBox[{"m", "+", "1"}]], " ", 
        RowBox[{"Gamma", "[", 
         RowBox[{"n", ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", "x"}]}]}], "]"}]}], 
       RowBox[{"d", " ", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}]]}]}]}]}], "+", 
  RowBox[{
   FractionBox["b", 
    RowBox[{"d", " ", 
     RowBox[{"(", 
      RowBox[{"m", "+", "1"}], ")"}]}]], 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], 
        RowBox[{"m", "+", "1"}]], " ", 
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", "x"}]}], ")"}], 
        RowBox[{"n", "-", "1"}]]}], 
      SuperscriptBox["\[ExponentialE]", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944393519535737`*^9, {3.4944398753843064`*^9, 
   3.494439898614339*^9}, {3.5487729084881315`*^9, 3.548772951029406*^9}, {
   3.548773033491151*^9, 3.548773102989273*^9}, {3.548773240924715*^9, 
   3.5487732690827646`*^9}, {3.7093101834524245`*^9, 3.7093102025425167`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", "x_"}]}]}], "]"}]}], ",", "x_Symbol"}], "]"}], 
  " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{"Block", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"$UseGamma", "=", "True"}], "}"}], ",", "\n", "    ", 
     RowBox[{
      RowBox[{
       RowBox[{
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", "*", "x"}]}], ")"}], "^", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], "*", 
       RowBox[{
        RowBox[{"Gamma", "[", 
         RowBox[{"n", ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", "x"}]}]}], "]"}], "/", 
        RowBox[{"(", 
         RowBox[{"d", "*", 
          RowBox[{"(", 
           RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}]}], " ", "+", " ", 
      "\[IndentingNewLine]", "    ", 
      RowBox[{
       RowBox[{"b", "/", 
        RowBox[{"(", 
         RowBox[{"d", "*", 
          RowBox[{"(", 
           RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}], "*", 
       RowBox[{"Int", "[", 
        RowBox[{
         RowBox[{
          RowBox[{
           RowBox[{"(", 
            RowBox[{"c", "+", 
             RowBox[{"d", "*", "x"}]}], ")"}], "^", 
           RowBox[{"(", 
            RowBox[{"m", "+", "1"}], ")"}]}], "*", 
          RowBox[{
           RowBox[{
            RowBox[{"(", 
             RowBox[{"a", "+", 
              RowBox[{"b", "*", "x"}]}], ")"}], "^", 
            RowBox[{"(", 
             RowBox[{"n", "-", "1"}], ")"}]}], "/", 
           RowBox[{"E", "^", 
            RowBox[{"(", 
             RowBox[{"a", "+", 
              RowBox[{"b", "*", "x"}]}], ")"}]}]}]}], ",", "x"}], "]"}]}]}]}],
     "]"}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "m", ",", "n"}], "}"}],
       ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"IGtQ", "[", 
       RowBox[{"m", ",", "0"}], "]"}], " ", "||", " ", 
      RowBox[{"IGtQ", "[", 
       RowBox[{"n", ",", "0"}], "]"}], " ", "||", " ", 
      RowBox[{"IntegersQ", "[", 
       RowBox[{"m", ",", "n"}], "]"}]}], ")"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{"m", ",", 
      RowBox[{"-", "1"}]}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, {3.494439298133498*^9, 
   3.4944393224735327`*^9}, 3.548773313979644*^9, {3.548774218001231*^9, 
   3.548774225130444*^9}, {3.548779276211317*^9, 3.548779290343337*^9}, 
   3.5487837127235365`*^9, {3.5488095573285336`*^9, 3.5488095656285453`*^9}, {
   3.552514591145564*^9, 3.5525145954255695`*^9}, {3.552514725435752*^9, 
   3.5525147288657565`*^9}, {3.6144377669286184`*^9, 
   3.6144378180275407`*^9}, {3.6144389616939545`*^9, 3.614438967274274*^9}, 
   3.7061416389083166`*^9, {3.7093085949753904`*^9, 3.7093086239820495`*^9}, {
   3.7093102455339756`*^9, 3.7093103001330986`*^9}, 3.7093253569351254`*^9, 
   3.7210770537283845`*^9, 3.7210774879292192`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["U:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.574964815567827*^9, 3.5749648195978327`*^9}, 
   3.574965002088088*^9, {3.5749650378381386`*^9, 3.5749650380781384`*^9}, {
   3.5749850938482723`*^9, 3.574985134328329*^9}, {3.7093086668295*^9, 
   3.7093086793042135`*^9}, 3.7093101790071707`*^9, {3.709322481818415*^9, 
   3.7093224877484255`*^9}, {3.7210768545959945`*^9, 3.7210768560430775`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5749851409783382`*^9, 
   3.5749851409783382`*^9}, {3.709308692338058*^9, 3.709308692338058*^9}, 
   3.7093225039256544`*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
   RowBox[{"Gamma", "[", 
    RowBox[{"n", ",", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "x"}]}]}], "]"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     RowBox[{"\[Integral]", 
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
       RowBox[{"Gamma", "[", 
        RowBox[{"n", ",", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", "x"}]}]}], "]"}], 
       RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944393519535737`*^9, {3.4944398753843064`*^9, 
   3.494439898614339*^9}, {3.5487729084881315`*^9, 3.548772951029406*^9}, {
   3.548773033491151*^9, 3.548773102989273*^9}, {3.548773240924715*^9, 
   3.5487732690827646`*^9}, {3.7093101834524245`*^9, 
   3.7093102025425167`*^9}, {3.7093225098536644`*^9, 3.709322525484892*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"Gamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", "x_"}]}]}], "]"}]}], ",", "x_Symbol"}], "]"}], 
  " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{"Unintegrable", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"c", "+", 
         RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
      RowBox[{"Gamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", "x"}]}]}], "]"}]}], ",", "x"}], "]"}], " ", "/;", 
   "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "m", ",", "n"}], "}"}], 
     ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, {3.494439298133498*^9, 
   3.4944393224735327`*^9}, 3.548773313979644*^9, {3.548774218001231*^9, 
   3.548774225130444*^9}, {3.548779276211317*^9, 3.548779290343337*^9}, 
   3.5487837127235365`*^9, {3.5488095573285336`*^9, 3.5488095656285453`*^9}, {
   3.552514591145564*^9, 3.5525145954255695`*^9}, {3.552514725435752*^9, 
   3.5525147288657565`*^9}, {3.6144377669286184`*^9, 
   3.6144378180275407`*^9}, {3.6144389616939545`*^9, 3.614438967274274*^9}, 
   3.7061416389083166`*^9, {3.7093085949753904`*^9, 3.7093086239820495`*^9}, {
   3.7093102455339756`*^9, 3.7093103001330986`*^9}, {3.7093225361875105`*^9, 
   3.709322586143799*^9}, 3.709325364460361*^9, 3.7123388616018033`*^9, 
   3.715550785496291*^9, 3.7210770587036686`*^9, 3.7210774945125957`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["2.",
   FontFamily->"Arial"]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{"u", " ", 
    RowBox[{"LogGamma", "[", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "x"}]}], "]"}], 
    RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9}, 
   3.574964614077545*^9, {3.5749651212282553`*^9, 3.5749651249582605`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     RowBox[{"LogGamma", "[", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749648252378407`*^9, 
   3.5749648279878445`*^9}, {3.574965127558264*^9, 3.5749651277682643`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Primitive rule", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.494439616563944*^9, 3.4944396188039474`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    FractionBox[
     RowBox[{"\[PartialD]", 
      TemplateBox[{RowBox[{"-", "2"}],"z"},
       "PolyGamma2"]}], 
     RowBox[{"\[PartialD]", "z"}],
     MultilineFunction->None], "\[LongEqual]", 
    TemplateBox[{"z"},
     "LogGamma"]}], TraditionalForm]]]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.494439616563944*^9, 3.494439653543996*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494439179913333*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   RowBox[{"LogGamma", "[", 
    RowBox[{"a", "+", 
     RowBox[{"b", " ", "x"}]}], "]"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox[
      RowBox[{"PolyGamma", "[", 
       RowBox[{
        RowBox[{"-", "2"}], ",", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "x"}]}]}], "]"}], "b"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944391698033185`*^9, 3.4944392275633993`*^9, 
   3.494439676964029*^9, 3.494439726504098*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{"LogGamma", "[", 
     RowBox[{"a_.", "+", 
      RowBox[{"b_.", "*", "x_"}]}], "]"}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"PolyGamma", "[", 
     RowBox[{
      RowBox[{"-", "2"}], ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", "*", "x"}]}]}], "]"}], "/", "b"}], " ", "/;", "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "b"}], "}"}], ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.4944391618433075`*^9, 
   3.494439665674013*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["2.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"LogGamma", "[", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9}, 
   3.574964614077545*^9, {3.5749651212282553`*^9, 3.5749651249582605`*^9}, {
   3.709323870106758*^9, 3.709323880652377*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"LogGamma", "[", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"m", "\[Element]", 
       SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749648344578533`*^9, 3.574964838117859*^9}, {
   3.5749651421482844`*^9, 3.574965142408285*^9}, 3.7093104695867906`*^9, {
   3.7093238843515835`*^9, 3.7093238866759872`*^9}, {3.7093241462062464`*^9, 
   3.7093241462062464`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by parts", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.494439616563944*^9, 3.4944396188039474`*^9}, {
   3.4944398027942047`*^9, 3.49443980647421*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"m", "\[Element]", 
   SuperscriptBox["\[DoubleStruckCapitalZ]", "+"]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.7093241569716663`*^9, 3.7093241569716663`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    RowBox[{"LogGamma", "[", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "x"}]}], "]"}], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
        RowBox[{"PolyGamma", "[", 
         RowBox[{
          RowBox[{"-", "2"}], ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", "x"}]}]}], "]"}]}], "b"]}]}]}]}], "-", 
  RowBox[{
   FractionBox[
    RowBox[{"d", " ", "m"}], "b"], 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], 
      RowBox[{"m", "-", "1"}]], " ", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{
       RowBox[{"-", "2"}], ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944391698033185`*^9, 3.4944392275633993`*^9, 
   3.494439676964029*^9, 3.49443973512411*^9, 3.4944397853241806`*^9, {
   3.7093104720359306`*^9, 3.709310480207398*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"LogGamma", "[", 
      RowBox[{"a_.", "+", 
       RowBox[{"b_.", "*", "x_"}]}], "]"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
     RowBox[{
      RowBox[{"PolyGamma", "[", 
       RowBox[{
        RowBox[{"-", "2"}], ",", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", "x"}]}]}], "]"}], "/", "b"}]}], " ", "-", "\n", 
    "  ", 
    RowBox[{"d", "*", 
     RowBox[{"m", "/", "b"}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"m", "-", "1"}], ")"}]}], "*", 
        RowBox[{"PolyGamma", "[", 
         RowBox[{
          RowBox[{"-", "2"}], ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", "x"}]}]}], "]"}]}], ",", "x"}], "]"}]}]}], " ", "/;",
    "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"a", ",", "b", ",", "c", ",", "d"}], "}"}], ",", "x"}], "]"}], 
    " ", "&&", " ", 
    RowBox[{"IGtQ", "[", 
     RowBox[{"m", ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.4944391618433075`*^9, 
   3.494439665674013*^9, 3.494439747274127*^9, {3.5488142770021076`*^9, 
   3.548814301642143*^9}, {3.709308722760112*^9, 3.7093087321981277`*^9}, {
   3.7093105364567137`*^9, 3.7093105559733477`*^9}, 3.709324153429059*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["U:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"LogGamma", "[", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.574964815567827*^9, 3.5749648195978327`*^9}, 
   3.574965002088088*^9, {3.5749650378381386`*^9, 3.5749650380781384`*^9}, {
   3.5749850938482723`*^9, 3.574985134328329*^9}, {3.7093086668295*^9, 
   3.7093086793042135`*^9}, 3.7093101790071707`*^9, {3.709322481818415*^9, 
   3.7093224877484255`*^9}, {3.7093239292962627`*^9, 
   3.7093239409962835`*^9}, {3.721076861741403*^9, 3.7210768634435005`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5749851409783382`*^9, 
   3.5749851409783382`*^9}, {3.709308692338058*^9, 3.709308692338058*^9}, 
   3.7093225039256544`*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
   RowBox[{"LogGamma", "[", 
    RowBox[{"a", "+", 
     RowBox[{"b", " ", "x"}]}], "]"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     RowBox[{"\[Integral]", 
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
       RowBox[{"LogGamma", "[", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "x"}]}], "]"}], 
       RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944393519535737`*^9, {3.4944398753843064`*^9, 
   3.494439898614339*^9}, {3.5487729084881315`*^9, 3.548772951029406*^9}, {
   3.548773033491151*^9, 3.548773102989273*^9}, {3.548773240924715*^9, 
   3.5487732690827646`*^9}, {3.7093101834524245`*^9, 
   3.7093102025425167`*^9}, {3.7093225098536644`*^9, 3.709322525484892*^9}, {
   3.709323930076264*^9, 3.7093239473786945`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"LogGamma", "[", 
      RowBox[{"a_.", "+", 
       RowBox[{"b_.", "*", "x_"}]}], "]"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{"Unintegrable", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"c", "+", 
         RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
      RowBox[{"LogGamma", "[", 
       RowBox[{"a", "+", 
        RowBox[{"b", "*", "x"}]}], "]"}]}], ",", "x"}], "]"}], " ", "/;", 
   "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "m"}], "}"}], ",", 
     "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, {3.494439298133498*^9, 
   3.4944393224735327`*^9}, 3.548773313979644*^9, {3.548774218001231*^9, 
   3.548774225130444*^9}, {3.548779276211317*^9, 3.548779290343337*^9}, 
   3.5487837127235365`*^9, {3.5488095573285336`*^9, 3.5488095656285453`*^9}, {
   3.552514591145564*^9, 3.5525145954255695`*^9}, {3.552514725435752*^9, 
   3.5525147288657565`*^9}, {3.6144377669286184`*^9, 
   3.6144378180275407`*^9}, {3.6144389616939545`*^9, 3.614438967274274*^9}, 
   3.7061416389083166`*^9, {3.7093085949753904`*^9, 3.7093086239820495`*^9}, {
   3.7093102455339756`*^9, 3.7093103001330986`*^9}, {3.7093225361875105`*^9, 
   3.709322586143799*^9}, {3.709323931605067*^9, 3.7093239625439215`*^9}, 
   3.709325030058429*^9, 3.712338861648806*^9, 3.715550787711418*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["3.",
   FontFamily->"Arial"]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{"u", " ", 
    RowBox[{"PolyGamma", "[", 
     RowBox[{"n", ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9}, 
   3.574964614077545*^9, {3.5749651212282553`*^9, 3.5749651249582605`*^9}, 
   3.5749651671783195`*^9, {3.5749652443784275`*^9, 3.5749652445784273`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     RowBox[{"PolyGamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.574964845647869*^9, 3.574964848247873*^9}, {
   3.574965170988325*^9, 3.574965171198325*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Primitive rule", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.494439616563944*^9, 3.4944396188039474`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    FractionBox[
     RowBox[{"\[PartialD]", 
      TemplateBox[{"n","z"},
       "PolyGamma2"]}], 
     RowBox[{"\[PartialD]", "z"}],
     MultilineFunction->None], "\[LongEqual]", 
    TemplateBox[{RowBox[{"n", "+", "1"}],"z"},
     "PolyGamma2"]}], TraditionalForm]]]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.494439616563944*^9, 3.494439653543996*^9}, 
   3.494440010294495*^9}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494439179913333*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   RowBox[{"PolyGamma", "[", 
    RowBox[{"n", ",", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "x"}]}]}], "]"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox[
      RowBox[{"PolyGamma", "[", 
       RowBox[{
        RowBox[{"n", "-", "1"}], ",", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "x"}]}]}], "]"}], "b"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944391698033185`*^9, 3.4944392275633993`*^9, 
   3.494439676964029*^9, 3.494439726504098*^9, {3.4944400211445103`*^9, 
   3.494440046474546*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{"PolyGamma", "[", 
     RowBox[{"n_", ",", 
      RowBox[{"a_.", "+", 
       RowBox[{"b_.", "*", "x_"}]}]}], "]"}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"PolyGamma", "[", 
     RowBox[{
      RowBox[{"n", "-", "1"}], ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", "*", "x"}]}]}], "]"}], "/", "b"}], " ", "/;", "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "b", ",", "n"}], "}"}], ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.4944391618433075`*^9, 
   3.494439665674013*^9, 3.4944400286445208`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["2.",
    FontFamily->"Arial"]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.5749614988631835`*^9}, 
   3.574964614077545*^9, {3.5749651212282553`*^9, 3.5749651249582605`*^9}, 
   3.5749651671783195`*^9, {3.5749651988783636`*^9, 3.574965203868371*^9}, 
   3.709310659895933*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"m", ">", "0"}]]]], "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749648548478823`*^9, 
   3.5749648584878874`*^9}, {3.574965177718334*^9, 3.574965207688376*^9}, 
   3.709310661861536*^9, {3.7093248695102463`*^9, 3.7093248695102463`*^9}, {
   3.7093255459128013`*^9, 3.709325546698846*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by parts", "Subsubsection",
 CellChangeTimes->{3.4941724567488375`*^9}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"m", ">", "0"}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944394162836637`*^9, 
   3.4944394240236745`*^9}, {3.49444016365471*^9, 3.494440166564714*^9}, {
   3.7093248727854333`*^9, 3.709324872786433*^9}, {3.7093255526131845`*^9, 
   3.70932555393426*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    RowBox[{"PolyGamma", "[", 
     RowBox[{"n", ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
        RowBox[{"PolyGamma", "[", 
         RowBox[{
          RowBox[{"n", "-", "1"}], ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", "x"}]}]}], "]"}]}], "b"]}]}]}]}], "-", 
  RowBox[{
   FractionBox[
    RowBox[{"d", " ", "m"}], "b"], 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], 
      RowBox[{"m", "-", "1"}]], " ", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{
       RowBox[{"n", "-", "1"}], ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944391698033185`*^9, {3.4944393463335657`*^9, 
   3.4944393633235893`*^9}, 3.4944394680437365`*^9, 3.4944402029647655`*^9, 
   3.4944402360048113`*^9, {3.7093106641879406`*^9, 3.709310673081956*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", "x_"}]}]}], "]"}]}], ",", "x_Symbol"}], "]"}], 
  " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
     RowBox[{
      RowBox[{"PolyGamma", "[", 
       RowBox[{
        RowBox[{"n", "-", "1"}], ",", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", "x"}]}]}], "]"}], "/", "b"}]}], " ", "-", " ", 
    RowBox[{"d", "*", 
     RowBox[{"m", "/", "b"}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"m", "-", "1"}], ")"}]}], "*", 
        RowBox[{"PolyGamma", "[", 
         RowBox[{
          RowBox[{"n", "-", "1"}], ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", "x"}]}]}], "]"}]}], ",", "x"}], "]"}]}]}], " ", "/;",
    "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "n"}], "}"}], ",", 
      "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"GtQ", "[", 
     RowBox[{"m", ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.4944391618433075`*^9, 
   3.494439288643485*^9, 3.494440114764642*^9, {3.5488142804021125`*^9, 
   3.548814306232149*^9}, {3.7093087386097393`*^9, 3.709308744026949*^9}, {
   3.7093106062278385`*^9, 3.709310640300298*^9}, {3.7093107078428173`*^9, 
   3.709310708264018*^9}, {3.7093248758906107`*^9, 3.7093248949236994`*^9}, 
   3.709325557345455*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"m", "<", 
       RowBox[{"-", "1"}]}]]]], "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.574964864687896*^9, 3.574964868357901*^9}, {
   3.5749652087983775`*^9, 3.574965209728379*^9}, 3.7093106808673697`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Inverted integration by parts", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.4944401273646593`*^9, 3.4944401302046633`*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"m", "<", 
   RowBox[{"-", "1"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944394162836637`*^9, 
   3.4944394240236745`*^9}, {3.49444016365471*^9, 3.494440185694741*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"c", "+", 
       RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
    RowBox[{"PolyGamma", "[", 
     RowBox[{"n", ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", " ", "x"}]}], ")"}], 
         RowBox[{"m", "+", "1"}]], " ", 
        RowBox[{"PolyGamma", "[", 
         RowBox[{"n", ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", "x"}]}]}], "]"}]}], 
       RowBox[{"d", " ", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}]]}]}]}]}], "-", 
  RowBox[{
   FractionBox["b", 
    RowBox[{"d", " ", 
     RowBox[{"(", 
      RowBox[{"m", "+", "1"}], ")"}]}]], 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], 
      RowBox[{"m", "+", "1"}]], " ", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{
       RowBox[{"n", "+", "1"}], ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9}, 
   3.4944395044737873`*^9, 3.494440205154768*^9, 3.494440258444843*^9, {
   3.7093106830201735`*^9, 3.7093106989186015`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", "x_"}]}]}], "]"}]}], ",", "x_Symbol"}], "]"}], 
  " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", "*", "x"}]}], ")"}], "^", 
      RowBox[{"(", 
       RowBox[{"m", "+", "1"}], ")"}]}], "*", 
     RowBox[{
      RowBox[{"PolyGamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", "x"}]}]}], "]"}], "/", 
      RowBox[{"(", 
       RowBox[{"d", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}]}], " ", "-", "\n", "  ", 
    RowBox[{
     RowBox[{"b", "/", 
      RowBox[{"(", 
       RowBox[{"d", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "+", 
           RowBox[{"d", "*", "x"}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"m", "+", "1"}], ")"}]}], "*", 
        RowBox[{"PolyGamma", "[", 
         RowBox[{
          RowBox[{"n", "+", "1"}], ",", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", "x"}]}]}], "]"}]}], ",", "x"}], "]"}]}]}], " ", "/;",
    "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "n"}], "}"}], ",", 
      "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"LtQ", "[", 
     RowBox[{"m", ",", 
      RowBox[{"-", "1"}]}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494439298133498*^9, 
   3.494440123504654*^9, {3.548814283252116*^9, 3.5488143108021555`*^9}, {
   3.709308749533759*^9, 3.7093087561793704`*^9}, {3.709310615977855*^9, 
   3.7093106461367083`*^9}, {3.7093107187532454`*^9, 3.709310727974862*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t\t", 
  RowBox[{Cell[TextData[StyleBox["U:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", "+", 
        RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{"n", ",", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.574964815567827*^9, 3.5749648195978327`*^9}, 
   3.574965002088088*^9, {3.5749650378381386`*^9, 3.5749650380781384`*^9}, {
   3.5749850938482723`*^9, 3.574985134328329*^9}, {3.7093086668295*^9, 
   3.7093086793042135`*^9}, 3.7093101790071707`*^9, {3.709322481818415*^9, 
   3.7093224877484255`*^9}, 3.7093256191209884`*^9, {3.7210768695048475`*^9, 
   3.7210768695048475`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944395435438423`*^9, 
   3.494439555963859*^9}, {3.5487742513228903`*^9, 3.5487742585925026`*^9}, {
   3.548779340309407*^9, 3.5487793720034513`*^9}, {3.5749851409783382`*^9, 
   3.5749851409783382`*^9}, {3.709308692338058*^9, 3.709308692338058*^9}, 
   3.7093225039256544`*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"c", "+", 
      RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
   RowBox[{"PolyGamma", "[", 
    RowBox[{"n", ",", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "x"}]}]}], "]"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     RowBox[{"\[Integral]", 
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         RowBox[{"c", "+", 
          RowBox[{"d", " ", "x"}]}], ")"}], "m"], " ", 
       RowBox[{"PolyGamma", "[", 
        RowBox[{"n", ",", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", "x"}]}]}], "]"}], 
       RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944393519535737`*^9, {3.4944398753843064`*^9, 
   3.494439898614339*^9}, {3.5487729084881315`*^9, 3.548772951029406*^9}, {
   3.548773033491151*^9, 3.548773102989273*^9}, {3.548773240924715*^9, 
   3.5487732690827646`*^9}, {3.7093101834524245`*^9, 
   3.7093102025425167`*^9}, {3.7093225098536644`*^9, 3.709322525484892*^9}, {
   3.7093256198540306`*^9, 3.709325620441064*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"c_.", "+", 
        RowBox[{"d_.", "*", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{"n_", ",", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", "x_"}]}]}], "]"}]}], ",", "x_Symbol"}], "]"}], 
  " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{"Unintegrable", "[", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"c", "+", 
         RowBox[{"d", "*", "x"}]}], ")"}], "^", "m"}], "*", 
      RowBox[{"PolyGamma", "[", 
       RowBox[{"n", ",", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", "x"}]}]}], "]"}]}], ",", "x"}], "]"}], " ", "/;", 
   "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "m", ",", "n"}], "}"}], 
     ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, {3.494439298133498*^9, 
   3.4944393224735327`*^9}, 3.548773313979644*^9, {3.548774218001231*^9, 
   3.548774225130444*^9}, {3.548779276211317*^9, 3.548779290343337*^9}, 
   3.5487837127235365`*^9, {3.5488095573285336`*^9, 3.5488095656285453`*^9}, {
   3.552514591145564*^9, 3.5525145954255695`*^9}, {3.552514725435752*^9, 
   3.5525147288657565`*^9}, {3.6144377669286184`*^9, 
   3.6144378180275407`*^9}, {3.6144389616939545`*^9, 3.614438967274274*^9}, 
   3.7061416389083166`*^9, {3.7093085949753904`*^9, 3.7093086239820495`*^9}, {
   3.7093102455339756`*^9, 3.7093103001330986`*^9}, {3.7093225361875105`*^9, 
   3.709322586143799*^9}, 3.709325364460361*^9, {3.7093256210670996`*^9, 
   3.709325622439178*^9}, 3.712338861853818*^9, 3.7155507895755243`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["4:",
   FontFamily->"Arial",
   FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"Gamma", "[", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}], "]"}], "n"], " ", 
    RowBox[{"PolyGamma", "[", 
     RowBox[{"0", ",", 
      RowBox[{"a", "+", 
       RowBox[{"b", " ", "x"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.574964876747913*^9, 3.5749648796779165`*^9}, {
   3.574965246738431*^9, 3.574965247598432*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Primitive rule", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.494439616563944*^9, 3.4944396188039474`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    FractionBox[
     RowBox[{"\[PartialD]", 
      SuperscriptBox[
       TemplateBox[{"z"},
        "Gamma"], "n"]}], 
     RowBox[{"\[PartialD]", "z"}],
     MultilineFunction->None], "=", 
    RowBox[{"n", " ", 
     TemplateBox[{"0","z"},
      "PolyGamma2"], " ", 
     SuperscriptBox[
      TemplateBox[{"z"},
       "Gamma"], "n"]}]}], TraditionalForm]]]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.494439616563944*^9, 3.494439653543996*^9}, 
   3.494440010294495*^9, {3.4944404285250807`*^9, 3.4944404324350863`*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944394162836637`*^9, 
   3.4944394240236745`*^9}, {3.49444016365471*^9, 3.494440166564714*^9}, 
   3.4944406703654194`*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"Gamma", "[", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "x"}]}], "]"}], "n"], " ", 
   RowBox[{"PolyGamma", "[", 
    RowBox[{"0", ",", 
     RowBox[{"a", "+", 
      RowBox[{"b", " ", "x"}]}]}], "]"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox[
      SuperscriptBox[
       RowBox[{"Gamma", "[", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "x"}]}], "]"}], "n"], 
      RowBox[{"b", " ", "n"}]]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, 3.4944391698033185`*^9, {3.4944393463335657`*^9, 
   3.4944393633235893`*^9}, 3.4944394680437365`*^9, 3.4944402029647655`*^9, 
   3.4944402360048113`*^9, 3.4944403327949467`*^9, 3.494440636445372*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"Gamma", "[", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", "x_"}]}], "]"}], "^", "n_."}], "*", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{"0", ",", 
       RowBox[{"a_.", "+", 
        RowBox[{"b_.", "*", "x_"}]}]}], "]"}]}], ",", "x_Symbol"}], "]"}], 
  " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"a", "+", 
       RowBox[{"b", "*", "x"}]}], "]"}], "^", "n"}], "/", 
    RowBox[{"(", 
     RowBox[{"b", "*", "n"}], ")"}]}], " ", "/;", "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "b", ",", "n"}], "}"}], ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.4944391618433075`*^9, 
   3.494439288643485*^9, 3.494440114764642*^9, 3.4944404118050575`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["5:",
   FontFamily->"Arial",
   FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", "x"}]}], ")"}], "!"}], ")"}], "n"], " ", 
    RowBox[{"PolyGamma", "[", 
     RowBox[{"0", ",", 
      RowBox[{"c", "+", 
       RowBox[{"b", " ", "x"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", "x"}], "  ", 
    StyleBox["when",
     FontFamily->"Arial",
     FontWeight->"Plain"], 
    StyleBox["  ",
     FontFamily->"Arial",
     FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
     RowBox[{"c", "\[Equal]", 
      RowBox[{"a", "+", "1"}]}]]]], "None"]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, {3.5749646587176075`*^9, 
   3.5749646751076307`*^9}, {3.5749648900479317`*^9, 
   3.5749648929879355`*^9}, {3.5749652551084423`*^9, 
   3.5749652564684443`*^9}, {3.709308819933083*^9, 3.7093088326491055`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Primitive rule", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.494439616563944*^9, 3.4944396188039474`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    FractionBox[
     RowBox[{"\[PartialD]", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"z", "!"}], ")"}], "n"]}], 
     RowBox[{"\[PartialD]", "z"}],
     MultilineFunction->None], "=", 
    RowBox[{"n", " ", 
     TemplateBox[{"0",RowBox[{"z", "+", "1"}]},
      "PolyGamma2"], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"z", "!"}], ")"}], "n"]}]}], TraditionalForm]]]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.494439616563944*^9, 3.494439653543996*^9}, 
   3.494440010294495*^9, {3.4944404285250807`*^9, 3.4944404324350863`*^9}, {
   3.494440468745137*^9, 3.49444047076514*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{"c", "\[Equal]", 
   RowBox[{"a", "+", "1"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, {3.4944394162836637`*^9, 
   3.4944394240236745`*^9}, {3.49444016365471*^9, 3.494440185694741*^9}, {
   3.7093088595786324`*^9, 3.7093088595786324`*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", "x"}]}], ")"}], "!"}], ")"}], "n"], " ", 
   RowBox[{"PolyGamma", "[", 
    RowBox[{"0", ",", 
     RowBox[{"c", "+", 
      RowBox[{"b", " ", "x"}]}]}], "]"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox[
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", "x"}]}], ")"}], "!"}], ")"}], "n"], 
      RowBox[{"b", " ", "n"}]]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4944393492135696`*^9, 3.494439366993595*^9}, 
   3.4944395044737873`*^9, 3.494440205154768*^9, 3.494440258444843*^9, 
   3.4944403380849543`*^9, 3.494440511185197*^9, {3.4944406020953236`*^9, 
   3.4944406032253256`*^9}, 3.494440650915392*^9, {3.7093088644146414`*^9, 
   3.709308865586643*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{"a_.", "+", 
          RowBox[{"b_.", "*", "x_"}]}], ")"}], "!"}], ")"}], "^", "n_."}], 
     "*", 
     RowBox[{"PolyGamma", "[", 
      RowBox[{"0", ",", 
       RowBox[{"c_.", "+", 
        RowBox[{"b_.", "*", "x_"}]}]}], "]"}]}], ",", "x_Symbol"}], "]"}], 
  " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", "*", "x"}]}], ")"}], "!"}], ")"}], "^", "n"}], "/", 
    RowBox[{"(", 
     RowBox[{"b", "*", "n"}], ")"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{"a", ",", "b", ",", "c", ",", "n"}], "}"}], ",", "x"}], "]"}], 
    " ", "&&", " ", 
    RowBox[{"EqQ", "[", 
     RowBox[{"c", ",", 
      RowBox[{"a", "+", "1"}]}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494439298133498*^9, 
   3.494440123504654*^9, 3.4944404180850663`*^9, {3.4944405683652763`*^9, 
   3.4944405833952975`*^9}, 3.7061415696082196`*^9, {3.7093087850902214`*^9, 
   3.709308792469034*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["6.",
   FontFamily->"Arial"]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{"u", " ", 
    RowBox[{"Gamma", "[", 
     RowBox[{"p", ",", 
      RowBox[{"d", " ", 
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", 
          RowBox[{"Log", "[", 
           RowBox[{"c", " ", 
            SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.574961534083233*^9}, {
   3.5749624306844883`*^9, 3.57496245323452*^9}, 3.7094967868381433`*^9, {
   3.709497602398791*^9, 3.7094976398549333`*^9}, {3.709497795152816*^9, 
   3.7094977978639708`*^9}, {3.7095251211553674`*^9, 3.709525121329377*^9}, {
   3.7230827547896986`*^9, 3.723082766892391*^9}, {3.723183118137722*^9, 
   3.723183118471741*^9}, 3.7231831759960313`*^9, 3.7231836103878775`*^9, {
   3.7233428169481936`*^9, 3.723342819751354*^9}, 3.7233428752985315`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"p", ",", 
       RowBox[{"d", " ", 
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", 
           RowBox[{"Log", "[", 
            RowBox[{"c", " ", 
             SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, 3.7094292446364384`*^9, 
   3.709496691964717*^9, 3.709496740529495*^9, {3.723081681033283*^9, 
   3.723081692283927*^9}, {3.723082505411435*^9, 3.7230825282757425`*^9}, 
   3.7230827268050985`*^9, {3.7230827704735956`*^9, 3.7230827718686757`*^9}, 
   3.7231831773801107`*^9, 3.7231836110599155`*^9, 3.723342897269788*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by parts", "Subsubsection",
 CellChangeTimes->{3.4941724567488375`*^9}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    SubscriptBox["\[PartialD]", "x"], 
    RowBox[{"Gamma", "[", 
     RowBox[{"p", ",", 
      RowBox[{"d", " ", 
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", 
          RowBox[{"Log", "[", 
           RowBox[{"c", " ", 
            SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}]}], 
   "\[Equal]", 
   RowBox[{"-", 
    FractionBox[
     RowBox[{"b", " ", "d", " ", "n", " ", 
      SuperscriptBox["\[ExponentialE]", 
       RowBox[{"-", "a"}]], " ", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"d", " ", 
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", 
            RowBox[{"Log", "[", 
             RowBox[{"c", " ", 
              SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}], ")"}], 
       RowBox[{"p", "-", "1"}]]}], 
     RowBox[{"x", " ", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"c", " ", 
         SuperscriptBox["x", "n"]}], ")"}], 
       RowBox[{"b", " ", "d"}]]}]]}]}]],
  CellChangeTimes->{{3.723081786820334*^9, 3.723081824585494*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.7230818341590414`*^9, 3.7230818410724373`*^9}, {
   3.723081935599844*^9, 3.723081935599844*^9}, {3.723183178051149*^9, 
   3.723183178051149*^9}, {3.7231832457560215`*^9, 3.7231832457560215`*^9}, {
   3.7231833489449234`*^9, 3.7231833489449234`*^9}, {3.723183611713953*^9, 
   3.723183636590376*^9}, {3.7231836884773436`*^9, 3.7231836884773436`*^9}, {
   3.7233428984128532`*^9, 3.7233428996359234`*^9}, {3.7233430426931057`*^9, 
   3.7233430486704473`*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494271052164215*^9, {
   3.494271408804714*^9, 3.494271432974748*^9}, {3.574959991751074*^9, 
   3.574959994221077*^9}, {3.7230903530142927`*^9, 3.723090353195303*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    RowBox[{"Gamma", "[", 
     RowBox[{"p", ",", 
      RowBox[{"d", " ", 
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", 
          RowBox[{"Log", "[", 
           RowBox[{"c", " ", 
            SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", "x"}]}], " ", 
    RowBox[{"Gamma", "[", 
     RowBox[{"p", ",", 
      RowBox[{"d", " ", 
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", 
          RowBox[{"Log", "[", 
           RowBox[{"c", " ", 
            SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}]}]}], "+", 
  RowBox[{"b", " ", "d", " ", "n", " ", 
   SuperscriptBox["\[ExponentialE]", 
    RowBox[{
     RowBox[{"-", "a"}], " ", "d"}]], " ", 
   RowBox[{"\[Integral]", " ", 
    RowBox[{
     FractionBox[
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"d", " ", 
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", 
            RowBox[{"Log", "[", 
             RowBox[{"c", " ", 
              SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}], ")"}], 
       RowBox[{"p", "-", "1"}]], 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"c", " ", 
         SuperscriptBox["x", "n"]}], ")"}], 
       RowBox[{"b", " ", "d"}]]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.494271065404234*^9, 3.494271092894272*^9}, {
   3.4942713522746353`*^9, 3.494271366464655*^9}, 3.494275024729777*^9, {
   3.7094292468135633`*^9, 3.709429263753532*^9}, {3.723081710084945*^9, 
   3.72308175719864*^9}, {3.723081856148299*^9, 3.7230819091783323`*^9}, 
   3.723081945182392*^9, {3.7230820167834873`*^9, 3.7230820234358673`*^9}, {
   3.7230825485209007`*^9, 3.7230825772195425`*^9}, 3.7230831000334454`*^9, {
   3.723183178662184*^9, 3.7231831792142153`*^9}, {3.723183273965635*^9, 
   3.7231833006501613`*^9}, {3.723183371228198*^9, 3.723183380610735*^9}, {
   3.7231834313606377`*^9, 3.7231834488986406`*^9}, {3.723183506179917*^9, 
   3.723183517917588*^9}, {3.723183612268985*^9, 3.7231836131010323`*^9}, 
   3.7231837057283306`*^9, {3.723342900295961*^9, 3.723342901663039*^9}, {
   3.723343053855744*^9, 3.7233430606791344`*^9}, {3.7233496132459197`*^9, 
   3.7233496145709953`*^9}, 3.7233496513070965`*^9},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{"Gamma", "[", 
     RowBox[{"p_", ",", 
      RowBox[{"d_.", "*", 
       RowBox[{"(", 
        RowBox[{"a_.", "+", 
         RowBox[{"b_.", "*", 
          RowBox[{"Log", "[", 
           RowBox[{"c_.", "*", 
            RowBox[{"x_", "^", "n_."}]}], "]"}]}]}], ")"}]}]}], "]"}], ",", 
    "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"x", "*", 
     RowBox[{"Gamma", "[", 
      RowBox[{"p", ",", 
       RowBox[{"d", "*", 
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", "*", 
           RowBox[{"Log", "[", 
            RowBox[{"c", "*", 
             RowBox[{"x", "^", "n"}]}], "]"}]}]}], ")"}]}]}], "]"}]}], " ", 
    "+", " ", 
    RowBox[{"b", "*", "d", "*", "n", "*", 
     RowBox[{"E", "^", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "a"}], "*", "d"}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"d", "*", 
           RowBox[{"(", 
            RowBox[{"a", "+", 
             RowBox[{"b", "*", 
              RowBox[{"Log", "[", 
               RowBox[{"c", "*", 
                RowBox[{"x", "^", "n"}]}], "]"}]}]}], ")"}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"p", "-", "1"}], ")"}]}], "/", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "*", 
           RowBox[{"x", "^", "n"}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"b", "*", "d"}], ")"}]}]}], ",", "x"}], "]"}]}]}], " ", "/;",
    "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "n", ",", "p"}], "}"}], 
     ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.4942710790942526`*^9, 
   3.4942713187345886`*^9, 3.4942750371797943`*^9, 3.5488204945868177`*^9, 
   3.548820596626961*^9, 3.706141678638372*^9, {3.7094287716213837`*^9, 
   3.7094287721314125`*^9}, {3.7094292930112057`*^9, 3.709429371395689*^9}, {
   3.7230820310533037`*^9, 3.7230820319793563`*^9}, {3.7230821266387706`*^9, 
   3.723082158833612*^9}, 3.7230822088254714`*^9, {3.72308260462811*^9, 
   3.7230826351638565`*^9}, 3.7230826785733395`*^9, {3.723183179918256*^9, 
   3.723183180462287*^9}, 3.723183563824214*^9, {3.7231836141740937`*^9, 
   3.723183614774128*^9}, {3.723183713722788*^9, 3.723183760249449*^9}, 
   3.7233429472006435`*^9, {3.723342989565067*^9, 3.7233429965374656`*^9}, {
   3.7233430899288073`*^9, 3.7233430947370825`*^9}, {3.7233431435208726`*^9, 
   3.7233431563096037`*^9}, {3.7233496223364396`*^9, 3.723349623371499*^9}, 
   3.7233496559293613`*^9},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     FractionBox[
      RowBox[{"Gamma", "[", 
       RowBox[{"p", ",", 
        RowBox[{"d", " ", 
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", 
            RowBox[{"Log", "[", 
             RowBox[{"c", " ", 
              SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}], "x"], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, 3.7094292446364384`*^9, 
   3.709496691964717*^9, 3.709496740529495*^9, {3.723081681033283*^9, 
   3.723081692283927*^9}, {3.723082505411435*^9, 3.7230825282757425`*^9}, 
   3.7230827268050985`*^9, {3.7230827704735956`*^9, 3.7230827718686757`*^9}, {
   3.723082930805766*^9, 3.7230829462486496`*^9}, 3.7231831810123186`*^9, 
   3.7231836201324344`*^9, 3.723342902614094*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by substitution", "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.7230829601854467`*^9, 3.7230829628055964`*^9}}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   FractionBox[
    RowBox[{"F", "[", 
     RowBox[{"Log", "[", 
      RowBox[{"c", " ", 
       SuperscriptBox["x", "n"]}], "]"}], "]"}], "x"], "\[Equal]", 
   RowBox[{
    FractionBox["1", "n"], " ", 
    RowBox[{"Subst", "[", 
     RowBox[{
      RowBox[{"F", "[", "x", "]"}], ",", "x", ",", 
      RowBox[{"Log", "[", 
       RowBox[{"c", " ", 
        SuperscriptBox["x", "n"]}], "]"}]}], "]"}], " ", 
    RowBox[{
     SubscriptBox["\[PartialD]", "x"], 
     RowBox[{"Log", "[", 
      RowBox[{"c", " ", 
       SuperscriptBox["x", "n"]}], "]"}]}]}]}]]]
}], "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.495321989219356*^9, 3.4953835047314157`*^9, 3.4953841387223034`*^9, 
   3.495384868433325*^9, {3.4953876713659415`*^9, 3.495387673145944*^9}, {
   3.4953891516580143`*^9, 3.495389154318018*^9}, 3.4953892091980944`*^9, {
   3.4953892828081975`*^9, 3.495389298008219*^9}, {3.7151167463770504`*^9, 
   3.7151167463770504`*^9}, {3.723063218657098*^9, 3.723063218657098*^9}, {
   3.723071513604542*^9, 3.723071513604542*^9}, {3.723071765282937*^9, 
   3.723071765282937*^9}, {3.7230718882389693`*^9, 3.7230719475683627`*^9}, {
   3.723071995683115*^9, 3.723071995683115*^9}, {3.7230720643290415`*^9, 
   3.723072075416675*^9}, {3.723083069080675*^9, 3.723083069080675*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494271052164215*^9, {
   3.494271408804714*^9, 3.494271432974748*^9}, {3.574959991751074*^9, 
   3.574959994221077*^9}, {3.72308307178683*^9, 3.7230830719318385`*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   FractionBox[
    RowBox[{"Gamma", "[", 
     RowBox[{"p", ",", 
      RowBox[{"d", " ", 
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", 
          RowBox[{"Log", "[", 
           RowBox[{"c", " ", 
            SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}], "x"], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox["1", "n"]}]}], " ", 
   RowBox[{"Subst", "[", 
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"p", ",", 
       RowBox[{"d", " ", 
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", "x"}]}], ")"}]}]}], "]"}], ",", "x", ",", 
     RowBox[{"Log", "[", 
      RowBox[{"c", " ", 
       SuperscriptBox["x", "n"]}], "]"}]}], "]"}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.494271065404234*^9, 3.494271092894272*^9}, {
   3.4942713522746353`*^9, 3.494271366464655*^9}, 3.494275024729777*^9, {
   3.7094292468135633`*^9, 3.709429263753532*^9}, {3.723081710084945*^9, 
   3.72308175719864*^9}, {3.723081856148299*^9, 3.7230819091783323`*^9}, 
   3.723081945182392*^9, {3.7230820167834873`*^9, 3.7230820234358673`*^9}, {
   3.7230825485209007`*^9, 3.7230825772195425`*^9}, {3.7230830782642*^9, 
   3.723083091262944*^9}, {3.7230831267099714`*^9, 3.723083140730773*^9}, {
   3.7231831815723505`*^9, 3.723183182124382*^9}, {3.7231836217885294`*^9, 
   3.723183622396564*^9}, 3.7233429034231396`*^9, {3.723343175799719*^9, 
   3.7233431840751925`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"p_", ",", 
       RowBox[{"d_.", "*", 
        RowBox[{"(", 
         RowBox[{"a_.", "+", 
          RowBox[{"b_.", "*", 
           RowBox[{"Log", "[", 
            RowBox[{"c_.", "*", 
             RowBox[{"x_", "^", "n_."}]}], "]"}]}]}], ")"}]}]}], "]"}], "/", 
     "x_"}], ",", "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"1", "/", "n"}], "*", 
    RowBox[{"Subst", "[", 
     RowBox[{
      RowBox[{"Gamma", "[", 
       RowBox[{"p", ",", 
        RowBox[{"d", "*", 
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", "x"}]}], ")"}]}]}], "]"}], ",", "x", ",", 
      RowBox[{"Log", "[", 
       RowBox[{"c", "*", 
        RowBox[{"x", "^", "n"}]}], "]"}]}], "]"}]}], " ", "/;", "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"a", ",", "b", ",", "c", ",", "d", ",", "n", ",", "p"}], "}"}], 
     ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.4942710790942526`*^9, 
   3.4942713187345886`*^9, 3.4942750371797943`*^9, 3.5488204945868177`*^9, 
   3.548820596626961*^9, 3.706141678638372*^9, {3.7094287716213837`*^9, 
   3.7094287721314125`*^9}, {3.7094292930112057`*^9, 3.709429371395689*^9}, {
   3.7230820310533037`*^9, 3.7230820319793563`*^9}, {3.7230821266387706`*^9, 
   3.723082158833612*^9}, 3.7230822088254714`*^9, {3.72308260462811*^9, 
   3.7230826351638565`*^9}, 3.7230826785733395`*^9, {3.72308316585121*^9, 
   3.72308319434584*^9}, {3.7230857267446847`*^9, 3.72308573784832*^9}, {
   3.7231838041099577`*^9, 3.723183825084157*^9}, 3.7233429522389317`*^9, 
   3.723343010239249*^9, {3.7233431915096173`*^9, 3.7233431967699184`*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["3:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"e", " ", "x"}], ")"}], "m"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{"p", ",", 
       RowBox[{"d", " ", 
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", 
           RowBox[{"Log", "[", 
            RowBox[{"c", " ", 
             SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{"m", "\[NotEqual]", 
       RowBox[{"-", "1"}]}]]]], "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.5525328819892797`*^9, 
   3.5527731546608505`*^9, 3.553222366079949*^9, {3.574878892237171*^9, 
   3.574878896043578*^9}, {3.5748794303045163`*^9, 3.57487944361134*^9}, 
   3.574881171829175*^9, {3.574887120235253*^9, 3.574887131005268*^9}, 
   3.5748877279261036`*^9, {3.574888405747052*^9, 3.574888405747052*^9}, {
   3.574959730430708*^9, 3.574959734700714*^9}, {3.5749599853510647`*^9, 
   3.574959999411084*^9}, 3.574961543243246*^9, 3.7094292446364384`*^9, 
   3.709496691964717*^9, 3.709496740529495*^9, {3.723081681033283*^9, 
   3.723081692283927*^9}, {3.723082505411435*^9, 3.7230825306638794`*^9}, {
   3.723082773250755*^9, 3.723082774206809*^9}, {3.7230829324188585`*^9, 
   3.72308293261687*^9}, 3.7231831829234276`*^9, 3.7231836231156054`*^9, 
   3.7233428353602467`*^9, 3.723342905383252*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by parts", "Subsubsection",
 CellChangeTimes->{3.4941724567488375`*^9}],

Cell[TextData[{
 "Basis: ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    SubscriptBox["\[PartialD]", "x"], 
    RowBox[{"Gamma", "[", 
     RowBox[{"p", ",", 
      RowBox[{"d", " ", 
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", 
          RowBox[{"Log", "[", 
           RowBox[{"c", " ", 
            SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}]}], 
   "\[Equal]", 
   RowBox[{"-", 
    FractionBox[
     RowBox[{"b", " ", "d", " ", "n", " ", 
      SuperscriptBox["\[ExponentialE]", 
       RowBox[{
        RowBox[{"-", "a"}], " ", "d"}]], " ", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"d", " ", 
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", " ", 
            RowBox[{"Log", "[", 
             RowBox[{"c", " ", 
              SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}], ")"}], 
       RowBox[{
        RowBox[{"-", "1"}], "+", "p"}]]}], 
     RowBox[{"x", " ", 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"c", " ", 
         SuperscriptBox["x", "n"]}], ")"}], 
       RowBox[{"b", " ", "d"}]]}]]}]}]],
  CellChangeTimes->{{3.723081786820334*^9, 3.723081824585494*^9}}]
}], "Subsubsection",
 CellChangeTimes->{
  3.4941724567488375`*^9, {3.7230818341590414`*^9, 3.7230818410724373`*^9}, {
   3.723081935599844*^9, 3.723081935599844*^9}, {3.723183178051149*^9, 
   3.723183178051149*^9}, {3.7231832457560215`*^9, 3.7231832457560215`*^9}, {
   3.7231833489449234`*^9, 3.7231833489449234`*^9}, {3.723183611713953*^9, 
   3.723183636590376*^9}, {3.7231836884773436`*^9, 3.7231836884773436`*^9}, {
   3.723342906636324*^9, 3.7233429072833605`*^9}, {3.7233432067274876`*^9, 
   3.723343212079794*^9}, {3.7233515107834525`*^9, 3.7233515107834525`*^9}}],

Cell[TextData[{
 "Rule: ",
 "If  ",
 Cell[BoxData[
  RowBox[{"m", "\[NotEqual]", 
   RowBox[{"-", "1"}]}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494271052164215*^9, {
   3.494271408804714*^9, 3.494271432974748*^9}, {3.574959991751074*^9, 
   3.574959994221077*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"\[Integral]", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"e", " ", "x"}], ")"}], "m"], " ", 
    RowBox[{"Gamma", "[", 
     RowBox[{"p", ",", 
      RowBox[{"d", " ", 
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", 
          RowBox[{"Log", "[", 
           RowBox[{"c", " ", 
            SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}], 
    RowBox[{"\[DifferentialD]", 
     RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
      FractionBox[
       RowBox[{
        SuperscriptBox[
         RowBox[{"(", 
          RowBox[{"e", " ", "x"}], ")"}], 
         RowBox[{"m", "+", "1"}]], " ", 
        RowBox[{"Gamma", "[", 
         RowBox[{"p", ",", 
          RowBox[{"d", " ", 
           RowBox[{"(", 
            RowBox[{"a", "+", 
             RowBox[{"b", " ", 
              RowBox[{"Log", "[", 
               RowBox[{"c", " ", 
                SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}]}], 
       RowBox[{"e", " ", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}]]}]}]}]}], "+", 
  RowBox[{
   FractionBox[
    RowBox[{"b", " ", "d", " ", "n", " ", 
     SuperscriptBox["\[ExponentialE]", 
      RowBox[{
       RowBox[{"-", "a"}], " ", "d"}]], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"e", " ", "x"}], ")"}], 
      RowBox[{"b", " ", "d", " ", "n"}]]}], 
    RowBox[{
     RowBox[{"(", 
      RowBox[{"m", "+", "1"}], ")"}], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"c", " ", 
        SuperscriptBox["x", "n"]}], ")"}], 
      RowBox[{"b", " ", "d"}]]}]], " ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"e", " ", "x"}], ")"}], 
      RowBox[{"m", "-", 
       RowBox[{"b", " ", "d", " ", "n"}]}]], " ", 
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"d", " ", 
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", 
           RowBox[{"Log", "[", 
            RowBox[{"c", " ", 
             SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}], ")"}], 
      RowBox[{"p", "-", "1"}]], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.494271065404234*^9, 3.494271092894272*^9}, {
   3.4942713522746353`*^9, 3.494271366464655*^9}, 3.494275024729777*^9, {
   3.7094292468135633`*^9, 3.709429263753532*^9}, {3.723081710084945*^9, 
   3.72308175719864*^9}, {3.723081856148299*^9, 3.7230819091783323`*^9}, 
   3.723081945182392*^9, {3.7230820167834873`*^9, 3.7230820234358673`*^9}, 
   3.7230825350871325`*^9, 3.7230831072968607`*^9, {3.7231831841584983`*^9, 
   3.7231831847825336`*^9}, {3.723183624334675*^9, 3.723183624854705*^9}, {
   3.723183874136963*^9, 3.723183928045046*^9}, 3.7231839620109887`*^9, 
   3.7231839966739717`*^9, {3.7231841266144037`*^9, 3.723184164371563*^9}, {
   3.723184197022431*^9, 3.72318421326836*^9}, {3.72334283646731*^9, 
   3.7233428384484234`*^9}, {3.7233429078003902`*^9, 3.723342909616494*^9}, {
   3.723343216410042*^9, 3.723343226598624*^9}, {3.723349644969734*^9, 
   3.7233496461308002`*^9}, 3.723351546785512*^9, {3.7233516378937225`*^9, 
   3.723351661755088*^9}, 3.7233518212572107`*^9, {3.7233518525460005`*^9, 
   3.7233518598224163`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"e_.", "*", "x_"}], ")"}], "^", "m_."}], "*", 
     RowBox[{"Gamma", "[", 
      RowBox[{"p_", ",", 
       RowBox[{"d_.", "*", 
        RowBox[{"(", 
         RowBox[{"a_.", "+", 
          RowBox[{"b_.", "*", 
           RowBox[{"Log", "[", 
            RowBox[{"c_.", "*", 
             RowBox[{"x_", "^", "n_."}]}], "]"}]}]}], ")"}]}]}], "]"}]}], ",",
     "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"e", "*", "x"}], ")"}], "^", 
      RowBox[{"(", 
       RowBox[{"m", "+", "1"}], ")"}]}], "*", 
     RowBox[{
      RowBox[{"Gamma", "[", 
       RowBox[{"p", ",", 
        RowBox[{"d", "*", 
         RowBox[{"(", 
          RowBox[{"a", "+", 
           RowBox[{"b", "*", 
            RowBox[{"Log", "[", 
             RowBox[{"c", "*", 
              RowBox[{"x", "^", "n"}]}], "]"}]}]}], ")"}]}]}], "]"}], "/", 
      RowBox[{"(", 
       RowBox[{"e", "*", 
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}]}], ")"}]}]}], " ", "+", " ", 
    "\[IndentingNewLine]", "  ", 
    RowBox[{"b", "*", "d", "*", "n", "*", 
     RowBox[{"E", "^", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"-", "a"}], "*", "d"}], ")"}]}], "*", 
     RowBox[{
      RowBox[{
       RowBox[{"(", 
        RowBox[{"e", "*", "x"}], ")"}], "^", 
       RowBox[{"(", 
        RowBox[{"b", "*", "d", "*", "n"}], ")"}]}], "/", 
      RowBox[{"(", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{"m", "+", "1"}], ")"}], "*", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"c", "*", 
           RowBox[{"x", "^", "n"}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"b", "*", "d"}], ")"}]}]}], ")"}]}], "*", 
     RowBox[{"Int", "[", 
      RowBox[{
       RowBox[{
        RowBox[{
         RowBox[{"(", 
          RowBox[{"e", "*", "x"}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"m", "-", 
           RowBox[{"b", "*", "d", "*", "n"}]}], ")"}]}], "*", 
        RowBox[{
         RowBox[{"(", 
          RowBox[{"d", "*", 
           RowBox[{"(", 
            RowBox[{"a", "+", 
             RowBox[{"b", "*", 
              RowBox[{"Log", "[", 
               RowBox[{"c", "*", 
                RowBox[{"x", "^", "n"}]}], "]"}]}]}], ")"}]}], ")"}], "^", 
         RowBox[{"(", 
          RowBox[{"p", "-", "1"}], ")"}]}]}], ",", "x"}], "]"}]}]}], " ", "/;",
    "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "m", ",", "n", ",", 
        "p"}], "}"}], ",", "x"}], "]"}], " ", "&&", " ", 
    RowBox[{"NeQ", "[", 
     RowBox[{"m", ",", 
      RowBox[{"-", "1"}]}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.4942710790942526`*^9, 
   3.4942713187345886`*^9, 3.4942750371797943`*^9, 3.5488204945868177`*^9, 
   3.548820596626961*^9, 3.706141678638372*^9, {3.7094287716213837`*^9, 
   3.7094287721314125`*^9}, {3.7094292930112057`*^9, 3.709429371395689*^9}, {
   3.7230820310533037`*^9, 3.7230820319793563`*^9}, {3.7230821266387706`*^9, 
   3.723082158833612*^9}, 3.7230822088254714`*^9, {3.7231831854155703`*^9, 
   3.7231831860156045`*^9}, {3.7231836254877405`*^9, 
   3.7231836259677687`*^9}, {3.723184012069852*^9, 3.723184062017709*^9}, {
   3.723184270905657*^9, 3.7231843150821834`*^9}, 3.723184357181591*^9, {
   3.723342842186637*^9, 3.72334284468178*^9}, 3.7233429539380293`*^9, 
   3.7233430125383806`*^9, {3.723343097722253*^9, 3.7233430989383225`*^9}, {
   3.723343234145056*^9, 3.7233432694770765`*^9}, {3.7233496734393625`*^9, 
   3.7233496740043945`*^9}, {3.7233519045339737`*^9, 3.723351927616294*^9}, {
   3.723352043872944*^9, 3.7233520440499535`*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{Cell[TextData[StyleBox["7.",
   FontFamily->"Arial"]], "None"], "  ", 
  RowBox[{"\[Integral]", 
   RowBox[{"u", " ", 
    RowBox[{"Gamma", "[", 
     RowBox[{"p", ",", 
      RowBox[{"f", " ", 
       RowBox[{"(", 
        RowBox[{"a", "+", 
         RowBox[{"b", " ", 
          RowBox[{"Log", "[", 
           RowBox[{"c", " ", 
            SuperscriptBox[
             RowBox[{"(", 
              RowBox[{"d", "+", 
               RowBox[{"e", " ", "x"}]}], ")"}], "n"]}], "]"}]}]}], ")"}]}]}],
      "]"}], 
    RowBox[{"\[DifferentialD]", "x"}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193415004828687`*^9, 3.5193415113404875`*^9}, {
   3.5193513965602303`*^9, 3.519351397420232*^9}, {3.5194037839416766`*^9, 
   3.5194037847117205`*^9}, 3.52105258275843*^9, 3.5210531741726685`*^9, {
   3.521057461672199*^9, 3.5210574760242243`*^9}, {3.521059881564049*^9, 
   3.521059886088057*^9}, {3.5213902158895197`*^9, 3.521390217543123*^9}, {
   3.521390827878595*^9, 3.521390829376197*^9}, {3.5214688792561245`*^9, 
   3.521468879443325*^9}, {3.523320201951167*^9, 3.523320202247567*^9}, {
   3.5288596764894824`*^9, 3.5288596768094826`*^9}, {3.529196266260353*^9, 
   3.5291962665099535`*^9}, {3.5291997050223927`*^9, 
   3.5291997052563934`*^9}, {3.529201346457276*^9, 3.529201347252877*^9}, {
   3.529285041073242*^9, 3.5292850412632427`*^9}, {3.529285215443486*^9, 
   3.529285215443486*^9}, {3.5295115356999426`*^9, 3.5295115433443794`*^9}, {
   3.529521412127842*^9, 3.529521414647986*^9}, {3.529529429975619*^9, 
   3.529529432955224*^9}, 3.5340074894011116`*^9, 3.5415537711400824`*^9, {
   3.5415538454742126`*^9, 3.54155384951462*^9}, {3.5417297197063265`*^9, 
   3.5417297252443366`*^9}, {3.546297494083555*^9, 3.5462974982435613`*^9}, {
   3.5462989134855423`*^9, 3.5462989153655453`*^9}, 3.548544146508506*^9, 
   3.5488738855784535`*^9, {3.548885292189272*^9, 3.5488853011492844`*^9}, {
   3.5488862802206554`*^9, 3.5488862887906675`*^9}, {3.5488873344221315`*^9, 
   3.5488873405921392`*^9}, {3.548887765702735*^9, 3.548887777862752*^9}, {
   3.5494211382132635`*^9, 3.549421138353264*^9}, {3.5532230040978823`*^9, 
   3.5532230095731955`*^9}, 3.5589035854423523`*^9, 3.558904571603733*^9, {
   3.5729232863509145`*^9, 3.5729232864913144`*^9}, {3.574364442189604*^9, 
   3.574364468839641*^9}, 3.57487965984332*^9, {3.5748797599018955`*^9, 
   3.574879760588297*^9}, {3.57488723261541*^9, 3.5748872327854104`*^9}, 
   3.574887727776103*^9, {3.5749614948631783`*^9, 3.574961534083233*^9}, {
   3.5749624306844883`*^9, 3.57496245323452*^9}, 3.7094967868381433`*^9, {
   3.709497602398791*^9, 3.7094976398549333`*^9}, {3.709497795152816*^9, 
   3.7094977978639708`*^9}, {3.7095251211553674`*^9, 3.709525121329377*^9}, {
   3.7230827547896986`*^9, 3.723082766892391*^9}, {3.723183118137722*^9, 
   3.723183118471741*^9}, 3.7231831759960313`*^9, 3.7231836103878775`*^9, {
   3.7231847944075994`*^9, 3.723184808473404*^9}, 3.7231848552630796`*^9},
 FontSize->12,
 FontWeight->"Bold"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["1:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     RowBox[{"Gamma", "[", 
      RowBox[{"p", ",", 
       RowBox[{"f", " ", 
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", 
           RowBox[{"Log", "[", 
            RowBox[{"c", " ", 
             SuperscriptBox[
              RowBox[{"(", 
               RowBox[{"d", "+", 
                RowBox[{"e", " ", "x"}]}], ")"}], "n"]}], "]"}]}]}], 
         ")"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.550596464087599*^9, {
   3.5505965767870445`*^9, 3.5505965767870445`*^9}, 3.5505966202465305`*^9, {
   3.5506039138305197`*^9, 3.55060391400453*^9}, {3.5506040039986773`*^9, 
   3.550604005776779*^9}, {3.5506178888272758`*^9, 3.5506178900273447`*^9}, {
   3.5506192681491685`*^9, 3.550619293563622*^9}, {3.5506195305301757`*^9, 
   3.550619538392626*^9}, 3.5508113633281665`*^9, 3.550850564117722*^9, 
   3.55085086060068*^9, {3.5511408787654934`*^9, 3.5511409652134385`*^9}, {
   3.5511410229287395`*^9, 3.5511410229287395`*^9}, {3.551141425269752*^9, 
   3.5511414262078056`*^9}, 3.5705813877894106`*^9, {3.6218999397769117`*^9, 
   3.6218999422930555`*^9}, 3.6219787890043974`*^9, 3.7164165990485487`*^9, 
   3.716421987700762*^9, 3.716422185800092*^9, 3.7164222994325914`*^9, {
   3.7164227223747826`*^9, 3.716422724410899*^9}, {3.7164229455835495`*^9, 
   3.71642294577756*^9}, {3.7164432255844016`*^9, 3.716443227234496*^9}, {
   3.716514365051463*^9, 3.7165143652584753`*^9}, {3.717464935281926*^9, 
   3.71746493551694*^9}, {3.717465180474951*^9, 3.7174651904435205`*^9}, {
   3.7174652766104493`*^9, 3.7174652766104493`*^9}, {3.717465399277466*^9, 
   3.717465399277466*^9}, {3.7175157430038195`*^9, 3.717515743690221*^9}, 
   3.717516574157679*^9, {3.7175187680311327`*^9, 3.7175187682339334`*^9}, {
   3.717521523034529*^9, 3.717521540822547*^9}, {3.717526275752369*^9, 
   3.717526287229026*^9}, {3.717529985037528*^9, 3.717529986549615*^9}, 
   3.71753223013194*^9, {3.717532321253152*^9, 3.717532321253152*^9}, 
   3.7176042700099173`*^9, 3.717628070752305*^9, {3.7176281976295624`*^9, 
   3.7176282399589834`*^9}, {3.717628365227148*^9, 3.717628367097255*^9}, {
   3.7230835037565374`*^9, 3.7230835103659153`*^9}, {3.7230835438968334`*^9, 
   3.7230835509012337`*^9}, 3.723083676691428*^9, {3.723088819541583*^9, 
   3.7230888716605635`*^9}, {3.7230890116575713`*^9, 
   3.7230890118905845`*^9}, {3.7230909566188164`*^9, 3.723090957491866*^9}, {
   3.723092430167099*^9, 3.7230924482221317`*^9}, {3.7231846125381966`*^9, 
   3.723184616542426*^9}, 3.7231848115255785`*^9, {3.723343308234294*^9, 
   3.723343315014682*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by substitution", "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.4940010628073196`*^9, 3.4941723607787027`*^9, 3.4941724567488375`*^9, 
   3.494172696119172*^9, 3.4951614417343683`*^9, 3.495161553867365*^9, 
   3.49516358311173*^9, 3.4951639989000616`*^9, 3.4951647977306643`*^9, {
   3.7164202433349895`*^9, 3.71642025627973*^9}, {3.717465285489957*^9, 
   3.7174652861599956`*^9}}],

Cell["Rule:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494172783029294*^9, {
   3.495162076203082*^9, 3.495162102707529*^9}, {3.495162264932214*^9, 
   3.495162277318636*^9}, {3.4951637747120676`*^9, 3.495163775320469*^9}, {
   3.4951642488437004`*^9, 3.4951642509653044`*^9}, {3.495200116396866*^9, 
   3.4952001199668703`*^9}, {3.495201853289297*^9, 3.4952018648993134`*^9}, {
   3.495202034479551*^9, 3.495202075309608*^9}, {3.4952022052197895`*^9, 
   3.495202209799796*^9}, {3.49520236235001*^9, 3.495202364530013*^9}, {
   3.4952024627101507`*^9, 3.495202466110155*^9}, {3.570583225241983*^9, 
   3.570583225241983*^9}, {3.570628124196412*^9, 3.570628124196412*^9}, {
   3.6219006048689528`*^9, 3.6219006048689528`*^9}, {3.621906584467966*^9, 
   3.6219065844689665`*^9}, {3.716440871523757*^9, 3.716440871523757*^9}, {
   3.716515300587973*^9, 3.716515300587973*^9}, {3.7171098732254496`*^9, 
   3.7171098732254496`*^9}, {3.7176049808155727`*^9, 
   3.7176049808155727`*^9}, {3.7176285219861145`*^9, 
   3.7176285219861145`*^9}, {3.7176286914568076`*^9, 
   3.7176286914568076`*^9}, {3.7230889784136696`*^9, 
   3.7230889904933605`*^9}, {3.723090971349659*^9, 3.723090971349659*^9}, {
   3.7230924761647296`*^9, 3.7230924761647296`*^9}, 3.723184622966793*^9}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   RowBox[{"Gamma", "[", 
    RowBox[{"p", ",", 
     RowBox[{"f", " ", 
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Log", "[", 
          RowBox[{"c", " ", 
           SuperscriptBox[
            RowBox[{"(", 
             RowBox[{"d", "+", 
              RowBox[{"e", " ", "x"}]}], ")"}], "n"]}], "]"}]}]}], ")"}]}]}], 
    "]"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox["1", "e"]}]}], " ", 
   RowBox[{"Subst", "[", 
    RowBox[{
     RowBox[{"\[Integral]", 
      RowBox[{
       RowBox[{"Gamma", "[", 
        RowBox[{"p", ",", 
         RowBox[{"f", " ", 
          RowBox[{"(", 
           RowBox[{"a", "+", 
            RowBox[{"b", " ", 
             RowBox[{"Log", "[", 
              RowBox[{"c", " ", 
               SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}], 
       RowBox[{"\[DifferentialD]", "x"}]}]}], ",", "x", ",", 
     RowBox[{"d", "+", 
      RowBox[{"e", " ", "x"}]}]}], "]"}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4941727384192314`*^9, 3.494172760679263*^9}, {
   3.495161485476845*^9, 3.4951615041032777`*^9}, {3.495161845931078*^9, 
   3.4951618787379355`*^9}, {3.495163638398227*^9, 3.495163646853442*^9}, 
   3.49516370254554*^9, {3.495164038477331*^9, 3.4951640737177925`*^9}, 
   3.495199899876562*^9, 3.49519994111662*^9, {3.4952000649467936`*^9, 
   3.495200075966809*^9}, {3.5705598844879603`*^9, 3.5705598906079693`*^9}, {
   3.6218987652867346`*^9, 3.6218988171897035`*^9}, {3.621898877151133*^9, 
   3.6218989022125664`*^9}, {3.71641658841094*^9, 3.71641658893897*^9}, 
   3.716417539660348*^9, {3.7164175776505213`*^9, 3.7164176175118012`*^9}, {
   3.716417719489634*^9, 3.716417726642043*^9}, {3.716420061775605*^9, 
   3.716420134034738*^9}, {3.716420189208894*^9, 3.7164201896279173`*^9}, {
   3.716420227586089*^9, 3.7164202282441263`*^9}, {3.7164216796781435`*^9, 
   3.716421698824239*^9}, {3.7164219936731033`*^9, 3.7164220172984543`*^9}, {
   3.7164417918133945`*^9, 3.7164417936525*^9}, {3.716442737914509*^9, 
   3.7164427536484084`*^9}, {3.716442789584464*^9, 3.7164427900774918`*^9}, {
   3.7164429999524965`*^9, 3.716443000696539*^9}, {3.717465198046956*^9, 
   3.7174651995320406`*^9}, {3.7174655904954023`*^9, 
   3.7174656051562414`*^9}, {3.717465668689875*^9, 3.717465671856056*^9}, {
   3.717521550127079*^9, 3.717521574749487*^9}, {3.7175263483225203`*^9, 
   3.71752638154642*^9}, 3.7176280709353156`*^9, {3.717628161003467*^9, 
   3.717628186363918*^9}, 3.717628249115507*^9, 3.7230835651880507`*^9, {
   3.7230836044012938`*^9, 3.723083608193511*^9}, {3.723088881682137*^9, 
   3.7230888836072474`*^9}, {3.72318463130227*^9, 3.7231846381156597`*^9}, {
   3.72334332705437*^9, 3.7233433355038533`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{"Gamma", "[", 
     RowBox[{"p_", ",", 
      RowBox[{"f_.", "*", 
       RowBox[{"(", 
        RowBox[{"a_.", "+", 
         RowBox[{"b_.", "*", 
          RowBox[{"Log", "[", 
           RowBox[{"c_.", "*", 
            RowBox[{
             RowBox[{"(", 
              RowBox[{"d_", "+", 
               RowBox[{"e_.", "*", "x_"}]}], ")"}], "^", "n_."}]}], "]"}]}]}],
         ")"}]}]}], "]"}], ",", "x_Symbol"}], "]"}], " ", ":=", "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"1", "/", "e"}], "*", 
    RowBox[{"Subst", "[", 
     RowBox[{
      RowBox[{"Int", "[", 
       RowBox[{
        RowBox[{"Gamma", "[", 
         RowBox[{"p", ",", 
          RowBox[{"f", "*", 
           RowBox[{"(", 
            RowBox[{"a", "+", 
             RowBox[{"b", "*", 
              RowBox[{"Log", "[", 
               RowBox[{"c", "*", 
                RowBox[{"x", "^", "n"}]}], "]"}]}]}], ")"}]}]}], "]"}], ",", 
        "x"}], "]"}], ",", "x", ",", 
      RowBox[{"d", "+", 
       RowBox[{"e", "*", "x"}]}]}], "]"}]}], " ", "/;", "\n", 
   RowBox[{"FreeQ", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
      "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", "n", ",", 
       "p"}], "}"}], ",", "x"}], "]"}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494172717389202*^9, 
   3.495161451843186*^9, 3.4951616675135646`*^9, 3.495163619631394*^9, 
   3.495164018025695*^9, 3.495199875426528*^9, 3.495200523437435*^9, 
   3.495200639177597*^9, 3.495200753047757*^9, 3.4952010407881594`*^9, 
   3.4952013363585734`*^9, {3.4952570835699644`*^9, 3.495257087673199*^9}, {
   3.4952590667663965`*^9, 3.495259071698679*^9}, {3.495259155129451*^9, 
   3.4952591595397034`*^9}, {3.4952592284316435`*^9, 3.495259233363926*^9}, {
   3.5705608083492537`*^9, 3.570560819349269*^9}, {3.570560875679348*^9, 
   3.5705608813493557`*^9}, {3.6219001333179817`*^9, 3.62190014570369*^9}, {
   3.621900249252613*^9, 3.621900251408736*^9}, {3.6219002913290195`*^9, 
   3.6219002928911085`*^9}, {3.7164228363883038`*^9, 3.716422849596059*^9}, {
   3.7164229128336763`*^9, 3.7164229352129564`*^9}, 3.7164392531651926`*^9, 
   3.7164392890572453`*^9, 3.716503389679708*^9, 3.7165045499320707`*^9, 
   3.7174656625935264`*^9, {3.717465704865944*^9, 3.717465746281313*^9}, {
   3.7175151029350953`*^9, 3.717515109081506*^9}, {3.717518774473944*^9, 
   3.717518776689148*^9}, {3.7175216087704334`*^9, 3.717521685960848*^9}, {
   3.717521720553827*^9, 3.7175217400349407`*^9}, {3.717521794665066*^9, 
   3.7175218210705757`*^9}, 3.7175222782607255`*^9, {3.7175253204747305`*^9, 
   3.7175253245319624`*^9}, {3.7175260629631987`*^9, 
   3.7175260662003837`*^9}, {3.717526314639593*^9, 3.717526341518131*^9}, {
   3.717526392845066*^9, 3.717526436988591*^9}, {3.717526739113872*^9, 
   3.7175267421670465`*^9}, {3.7176282790592194`*^9, 3.717628331446216*^9}, {
   3.7230837061881156`*^9, 3.723083720287922*^9}, {3.723083756111971*^9, 
   3.723083762583341*^9}, {3.7230839975807824`*^9, 3.723084046227565*^9}, {
   3.72308524831032*^9, 3.723085280867182*^9}, {3.72308560357164*^9, 
   3.723085631512238*^9}, 3.7230888941038475`*^9, {3.7230892623589106`*^9, 
   3.723089270199359*^9}, 3.723089339570327*^9, 3.723090974041813*^9, 
   3.723092476818767*^9, {3.7231846586808357`*^9, 3.723184663401106*^9}, {
   3.723184703097377*^9, 3.723184736355279*^9}, {3.7233433541449194`*^9, 
   3.723343376764213*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"\t", 
  RowBox[{Cell[TextData[StyleBox["2:",
    FontFamily->"Arial",
    FontColor->RGBColor[1, 0, 0]]], "None"], "  ", 
   RowBox[{"\[Integral]", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"(", 
       RowBox[{"g", "+", 
        RowBox[{"h", " ", "x"}]}], ")"}], "m"], " ", 
     RowBox[{"Gamma", "[", 
      RowBox[{"p", ",", 
       RowBox[{"f", " ", 
        RowBox[{"(", 
         RowBox[{"a", "+", 
          RowBox[{"b", " ", 
           RowBox[{"Log", "[", 
            RowBox[{"c", " ", 
             SuperscriptBox[
              RowBox[{"(", 
               RowBox[{"d", "+", 
                RowBox[{"e", " ", "x"}]}], ")"}], "n"]}], "]"}]}]}], 
         ")"}]}]}], "]"}], 
     RowBox[{"\[DifferentialD]", "x"}], "  ", 
     StyleBox["when",
      FontFamily->"Arial",
      FontWeight->"Plain"], 
     StyleBox["  ",
      FontFamily->"Arial",
      FontWeight->"Plain"], Cell[TextData[Cell[BoxData[
      RowBox[{
       RowBox[{
        RowBox[{"e", " ", "g"}], "-", 
        RowBox[{"d", " ", "h"}]}], "\[Equal]", "0"}]]]], 
      "None"]}]}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{
  3.477935275784027*^9, {3.477935350060832*^9, 3.477935352384173*^9}, {
   3.477935393803731*^9, 3.477935400102789*^9}, {3.477935515378547*^9, 
   3.47793552034569*^9}, 3.478120029940968*^9, {3.479318860690858*^9, 
   3.47931886615872*^9}, {3.4794185033278093`*^9, 3.479418503918659*^9}, 
   3.4795122294117584`*^9, {3.479615913307893*^9, 3.4796159352794867`*^9}, 
   3.4796160386581373`*^9, 3.4796867204045024`*^9, {3.4928025699585147`*^9, 
   3.492802570130115*^9}, 3.492822274480175*^9, 3.492825822242408*^9, {
   3.4940970615585365`*^9, 3.494097062168537*^9}, {3.496441349250718*^9, 
   3.496441349830719*^9}, 3.4964414848609076`*^9, 3.496521708137803*^9, {
   3.4965218981240697`*^9, 3.49652189838407*^9}, 3.4965229070755*^9, 
   3.519247079685614*^9, {3.5193208582062006`*^9, 3.5193208612170057`*^9}, 
   3.5193325694253187`*^9, {3.5193422443984776`*^9, 3.5193422447260776`*^9}, 
   3.5193424488212357`*^9, {3.5193434708254333`*^9, 3.519343473103037*^9}, {
   3.5193435097631016`*^9, 3.51934353169674*^9}, {3.519345739319023*^9, 
   3.519345739599823*^9}, 3.5193462601259375`*^9, {3.5193475061060295`*^9, 
   3.5193475233908596`*^9}, {3.519351391880224*^9, 3.5193513931402254`*^9}, {
   3.519403773021052*^9, 3.5194037798564425`*^9}, 3.5194106793250647`*^9, {
   3.519537578341393*^9, 3.5195375792305946`*^9}, {3.5195376277778797`*^9, 
   3.51953767927357*^9}, {3.5197586260636144`*^9, 3.5197586260636144`*^9}, {
   3.51978053295651*^9, 3.519780542862528*^9}, {3.5197806526087203`*^9, 
   3.5197806657595434`*^9}, {3.5197807490168896`*^9, 3.5197807642581167`*^9}, 
   3.5197809004619555`*^9, {3.519782653514635*^9, 3.5197826623286505`*^9}, {
   3.5200207810164795`*^9, 3.520020785046485*^9}, {3.5213126833374643`*^9, 
   3.5213126833374643`*^9}, {3.521328235411002*^9, 3.521328235411002*^9}, {
   3.523315567893022*^9, 3.523315567893022*^9}, {3.5233157569332867`*^9, 
   3.5233157613032923`*^9}, {3.5323023198133526`*^9, 3.532302355723403*^9}, {
   3.5323026417638035`*^9, 3.5323026417638035`*^9}, {3.5323026980138817`*^9, 
   3.5323026980138817`*^9}, {3.532636287104328*^9, 3.532636287104328*^9}, {
   3.5330803686035347`*^9, 3.533080368863535*^9}, {3.533248909289961*^9, 
   3.5332489110299635`*^9}, {3.534962469565694*^9, 3.5349624910757236`*^9}, {
   3.5349625224057674`*^9, 3.534962522655768*^9}, {3.534965265519608*^9, 
   3.534965265519608*^9}, {3.5349668288168063`*^9, 3.5349668292668066`*^9}, {
   3.5349747358481646`*^9, 3.5349747358481646`*^9}, {3.534975457949176*^9, 
   3.534975457949176*^9}, {3.5368633963532934`*^9, 3.536863396571694*^9}, {
   3.541644322511304*^9, 3.5416443577513533`*^9}, {3.5416458269234104`*^9, 
   3.541645839633428*^9}, {3.5416460312236967`*^9, 3.541646031483697*^9}, 
   3.541646227783972*^9, {3.541647080335165*^9, 3.5416471077052035`*^9}, {
   3.541648997607849*^9, 3.5416489978278494`*^9}, 3.541649088407976*^9, {
   3.5417848711596775`*^9, 3.541784902429721*^9}, {3.541786145331461*^9, 
   3.5417861455914617`*^9}, {3.5417870037166634`*^9, 3.5417870039366636`*^9}, 
   3.5462979620242105`*^9, {3.546298264094633*^9, 3.546298265344635*^9}, {
   3.546299000235664*^9, 3.546299001505666*^9}, {3.546315119909564*^9, 
   3.5463151267659564`*^9}, {3.547227356319705*^9, 3.547227356319705*^9}, 
   3.547918559331044*^9, {3.547918610170952*^9, 3.5479186304411116`*^9}, 
   3.548537755517557*^9, 3.5485441629685287`*^9, 3.5485517520154133`*^9, {
   3.5488740727887154`*^9, 3.5488740727887154`*^9}, {3.548874206618903*^9, 
   3.548874206808903*^9}, 3.548877100795455*^9, 3.550596464087599*^9, {
   3.5505965767870445`*^9, 3.5505965767870445`*^9}, 3.5505966202465305`*^9, {
   3.5506039138305197`*^9, 3.55060391400453*^9}, {3.5506040039986773`*^9, 
   3.550604005776779*^9}, {3.5506178888272758`*^9, 3.5506178900273447`*^9}, {
   3.5506192681491685`*^9, 3.550619293563622*^9}, {3.5506195305301757`*^9, 
   3.550619538392626*^9}, 3.5508113633281665`*^9, 3.550850564117722*^9, 
   3.55085086060068*^9, {3.5511408787654934`*^9, 3.5511409652134385`*^9}, {
   3.5511410229287395`*^9, 3.5511410229287395`*^9}, {3.551141425269752*^9, 
   3.5511414262078056`*^9}, 3.5705813877894106`*^9, {3.6218999397769117`*^9, 
   3.6218999422930555`*^9}, 3.6219787890043974`*^9, 3.7164165990485487`*^9, 
   3.716421987700762*^9, 3.716422185800092*^9, 3.7164222994325914`*^9, {
   3.7164227223747826`*^9, 3.716422724410899*^9}, {3.7164229455835495`*^9, 
   3.71642294577756*^9}, {3.7164432255844016`*^9, 3.716443227234496*^9}, {
   3.716514365051463*^9, 3.7165143652584753`*^9}, {3.717464935281926*^9, 
   3.71746493551694*^9}, {3.717465180474951*^9, 3.7174651904435205`*^9}, {
   3.7174652766104493`*^9, 3.7174652766104493`*^9}, {3.717465399277466*^9, 
   3.717465399277466*^9}, {3.7175157430038195`*^9, 3.717515743690221*^9}, 
   3.717516574157679*^9, {3.7175187680311327`*^9, 3.7175187682339334`*^9}, {
   3.717521523034529*^9, 3.717521540822547*^9}, {3.717526275752369*^9, 
   3.717526287229026*^9}, {3.717529985037528*^9, 3.717529986549615*^9}, 
   3.71753223013194*^9, {3.717532321253152*^9, 3.717532321253152*^9}, 
   3.7176042700099173`*^9, 3.717628070752305*^9, {3.7176281976295624`*^9, 
   3.7176282399589834`*^9}, {3.717628365227148*^9, 3.717628367097255*^9}, {
   3.7230835037565374`*^9, 3.7230835103659153`*^9}, {3.7230835438968334`*^9, 
   3.7230835509012337`*^9}, 3.723083676691428*^9, {3.723083857947796*^9, 
   3.723083890977685*^9}, {3.723088938340378*^9, 3.7230889598106055`*^9}, {
   3.723089013167658*^9, 3.723089013351668*^9}, {3.723090983134333*^9, 
   3.723090983134333*^9}, {3.7230924774768047`*^9, 3.7230924774768047`*^9}, {
   3.723184813672701*^9, 3.723184821151129*^9}, 3.723185051620311*^9, {
   3.723343392249099*^9, 3.72334340852503*^9}, {3.7233434682544465`*^9, 
   3.7233435006322985`*^9}},
 FontSize->12,
 FontWeight->"Bold"],

Cell["Derivation: Integration by substitution", "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.4940010628073196`*^9, 3.4941723607787027`*^9, 3.4941724567488375`*^9, 
   3.494172696119172*^9, 3.4951614417343683`*^9, 3.495161553867365*^9, 
   3.49516358311173*^9, 3.4951639989000616`*^9, 3.4951647977306643`*^9, {
   3.7164202433349895`*^9, 3.71642025627973*^9}, {3.717465285489957*^9, 
   3.7174652861599956`*^9}}],

Cell[TextData[{
 "Basis:",
 " If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{"e", " ", "g"}], "-", 
    RowBox[{"d", " ", "h"}]}], "\[Equal]", "0"}]]],
 ", then",
 " ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"g", "+", 
       RowBox[{"h", " ", "x"}]}], ")"}], "m"], " ", 
    RowBox[{"F", "[", 
     RowBox[{"d", "+", 
      RowBox[{"e", " ", "x"}]}], "]"}]}], "\[Equal]", 
   RowBox[{
    FractionBox["1", "e"], " ", 
    RowBox[{"Subst", "[", 
     RowBox[{
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         FractionBox[
          RowBox[{"g", " ", "x"}], "d"], ")"}], "m"], " ", 
       RowBox[{"F", "[", "x", "]"}]}], ",", "x", ",", 
      RowBox[{"d", "+", 
       RowBox[{"e", " ", "x"}]}]}], "]"}], " ", 
    RowBox[{
     SubscriptBox["\[PartialD]", "x"], 
     RowBox[{"(", 
      RowBox[{"d", "+", 
       RowBox[{"e", " ", "x"}]}], ")"}]}]}]}]]]
}], "Subsubsection",
 CellChangeTimes->{
  3.479686820458373*^9, {3.490657960122612*^9, 3.49065796095366*^9}, 
   3.4940010628073196`*^9, 3.4941723607787027`*^9, 3.4941724567488375`*^9, 
   3.494172696119172*^9, 3.4951614417343683`*^9, 3.495161553867365*^9, 
   3.49516358311173*^9, 3.4951639989000616`*^9, {3.4951648090562844`*^9, 
   3.495164816591098*^9}, 3.4951998608665075`*^9, 3.4951999947566953`*^9, {
   3.5511408209821887`*^9, 3.5511408223792686`*^9}, {3.7164202833232765`*^9, 
   3.716420283324277*^9}, {3.716420431742766*^9, 3.716420431742766*^9}, {
   3.716443173263409*^9, 3.716443173263409*^9}, {3.7164440142795124`*^9, 
   3.7164440142795124`*^9}, {3.716444419537692*^9, 3.716444419537692*^9}, {
   3.7171028847619705`*^9, 3.717102886043173*^9}, {3.717105697839674*^9, 
   3.717105697839674*^9}, {3.717109462350363*^9, 3.7171095087800446`*^9}, {
   3.717109812189581*^9, 3.717109812189581*^9}, 3.7171099952100644`*^9, {
   3.7176049792014804`*^9, 3.7176049888680334`*^9}, {3.717605337698985*^9, 
   3.7176053526618414`*^9}, {3.7176285205240307`*^9, 
   3.7176285432913327`*^9}, {3.7176286898887177`*^9, 
   3.7176287116169605`*^9}, {3.717628811528675*^9, 3.71762883748516*^9}, {
   3.723083918902282*^9, 3.7230839217464447`*^9}, {3.7233434123962517`*^9, 
   3.7233434133113036`*^9}, {3.7233434757988777`*^9, 3.723343478316022*^9}}],

Cell[TextData[{
 "Rule: If  ",
 Cell[BoxData[
  RowBox[{
   RowBox[{
    RowBox[{"e", " ", "g"}], "-", 
    RowBox[{"d", " ", "h"}]}], "\[Equal]", "0"}]]],
 ", then"
}], "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.48070534977779*^9, {3.4928296953212276`*^9, 3.492829707041244*^9}, 
   3.4941724084687696`*^9, 3.4941724567488375`*^9, 3.494172783029294*^9, {
   3.495162076203082*^9, 3.495162102707529*^9}, {3.495162264932214*^9, 
   3.495162277318636*^9}, {3.4951637747120676`*^9, 3.495163775320469*^9}, {
   3.4951642488437004`*^9, 3.4951642509653044`*^9}, {3.495200116396866*^9, 
   3.4952001199668703`*^9}, {3.495201853289297*^9, 3.4952018648993134`*^9}, {
   3.495202034479551*^9, 3.495202075309608*^9}, {3.4952022052197895`*^9, 
   3.495202209799796*^9}, {3.49520236235001*^9, 3.495202364530013*^9}, {
   3.4952024627101507`*^9, 3.495202466110155*^9}, {3.570583225241983*^9, 
   3.570583225241983*^9}, {3.570628124196412*^9, 3.570628124196412*^9}, {
   3.6219006048689528`*^9, 3.6219006048689528`*^9}, {3.621906584467966*^9, 
   3.6219065844689665`*^9}, {3.716440871523757*^9, 3.716440871523757*^9}, {
   3.716515300587973*^9, 3.716515300587973*^9}, {3.7171098732254496`*^9, 
   3.7171098732254496`*^9}, {3.7176049808155727`*^9, 
   3.7176049808155727`*^9}, {3.7176285219861145`*^9, 
   3.7176285219861145`*^9}, {3.7176286914568076`*^9, 
   3.7176286914568076`*^9}, {3.7230889784136696`*^9, 3.72308897841467*^9}, {
   3.723090986351517*^9, 3.723090986351517*^9}, {3.7230924781248417`*^9, 
   3.7230924781248417`*^9}, {3.723184839423174*^9, 3.7231848396591873`*^9}, {
   3.723343414059346*^9, 3.723343414059346*^9}, {3.7233434825412636`*^9, 
   3.7233434825412636`*^9}}],

Cell[BoxData[
 RowBox[{"\[Integral]", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"g", "+", 
      RowBox[{"h", " ", "x"}]}], ")"}], "m"], " ", 
   RowBox[{"Gamma", "[", 
    RowBox[{"p", ",", 
     RowBox[{"f", " ", 
      RowBox[{"(", 
       RowBox[{"a", "+", 
        RowBox[{"b", " ", 
         RowBox[{"Log", "[", 
          RowBox[{"c", " ", 
           SuperscriptBox[
            RowBox[{"(", 
             RowBox[{"d", "+", 
              RowBox[{"e", " ", "x"}]}], ")"}], "n"]}], "]"}]}]}], ")"}]}]}], 
    "]"}], 
   RowBox[{"\[DifferentialD]", 
    RowBox[{"x", "  ", "\[LongRightArrow]", "  ", 
     FractionBox["1", "e"]}]}], " ", 
   RowBox[{"Subst", "[", 
    RowBox[{
     RowBox[{"\[Integral]", 
      RowBox[{
       SuperscriptBox[
        RowBox[{"(", 
         FractionBox[
          RowBox[{"g", " ", "x"}], "d"], ")"}], "m"], " ", 
       RowBox[{"Gamma", "[", 
        RowBox[{"p", ",", 
         RowBox[{"f", " ", 
          RowBox[{"(", 
           RowBox[{"a", "+", 
            RowBox[{"b", " ", 
             RowBox[{"Log", "[", 
              RowBox[{"c", " ", 
               SuperscriptBox["x", "n"]}], "]"}]}]}], ")"}]}]}], "]"}], 
       RowBox[{"\[DifferentialD]", "x"}]}]}], ",", "x", ",", 
     RowBox[{"d", "+", 
      RowBox[{"e", " ", "x"}]}]}], "]"}]}]}]], "Subsubtitle",
 CellDingbat->None,
 CellChangeTimes->{{3.4796579732027807`*^9, 3.479658002474872*^9}, {
   3.4796580599975853`*^9, 3.4796580609489536`*^9}, {3.479658295195784*^9, 
   3.47965829684816*^9}, 3.4796611938939075`*^9, 3.4796613924794593`*^9, {
   3.479661443342597*^9, 3.479661446647349*^9}, {3.479661576554146*^9, 
   3.479661589302477*^9}, 3.479661847553824*^9, {3.4796622294729967`*^9, 
   3.4796622450654173`*^9}, {3.479686720424531*^9, 3.4796867204345455`*^9}, {
   3.4796874646646957`*^9, 3.4796874745589232`*^9}, {3.492826573513727*^9, 
   3.492826578630536*^9}, {3.4928267070863624`*^9, 3.492826707460763*^9}, 
   3.4928267903497095`*^9, {3.492829799151373*^9, 3.4928299617216005`*^9}, {
   3.4940972077187414`*^9, 3.4940972347887793`*^9}, {3.49409966398218*^9, 
   3.494099664952181*^9}, {3.4940997156722527`*^9, 3.494099716112253*^9}, 
   3.4941724567488375`*^9, {3.4941727384192314`*^9, 3.494172760679263*^9}, {
   3.495161485476845*^9, 3.4951615041032777`*^9}, {3.495161845931078*^9, 
   3.4951618787379355`*^9}, {3.495163638398227*^9, 3.495163646853442*^9}, 
   3.49516370254554*^9, {3.495164038477331*^9, 3.4951640737177925`*^9}, 
   3.495199899876562*^9, 3.49519994111662*^9, {3.4952000649467936`*^9, 
   3.495200075966809*^9}, {3.5705598844879603`*^9, 3.5705598906079693`*^9}, {
   3.6218987652867346`*^9, 3.6218988171897035`*^9}, {3.621898877151133*^9, 
   3.6218989022125664`*^9}, {3.71641658841094*^9, 3.71641658893897*^9}, 
   3.716417539660348*^9, {3.7164175776505213`*^9, 3.7164176175118012`*^9}, {
   3.716417719489634*^9, 3.716417726642043*^9}, {3.716420061775605*^9, 
   3.716420134034738*^9}, {3.716420189208894*^9, 3.7164201896279173`*^9}, {
   3.716420227586089*^9, 3.7164202282441263`*^9}, {3.7164216796781435`*^9, 
   3.716421698824239*^9}, {3.7164219936731033`*^9, 3.7164220172984543`*^9}, {
   3.7164417918133945`*^9, 3.7164417936525*^9}, {3.716442737914509*^9, 
   3.7164427536484084`*^9}, {3.716442789584464*^9, 3.7164427900774918`*^9}, {
   3.7164429999524965`*^9, 3.716443000696539*^9}, {3.717465198046956*^9, 
   3.7174651995320406`*^9}, {3.7174655904954023`*^9, 
   3.7174656051562414`*^9}, {3.717465668689875*^9, 3.717465671856056*^9}, {
   3.717521550127079*^9, 3.717521574749487*^9}, {3.7175263483225203`*^9, 
   3.71752638154642*^9}, 3.7176280709353156`*^9, {3.717628161003467*^9, 
   3.717628186363918*^9}, 3.717628249115507*^9, 3.7230835651880507`*^9, {
   3.7230836044012938`*^9, 3.723083608193511*^9}, 3.7230838790910053`*^9, {
   3.723083961540721*^9, 3.7230839618837404`*^9}, {3.7231849180006685`*^9, 
   3.7231849218748903`*^9}, {3.723185017437356*^9, 3.7231850224596434`*^9}, 
   3.7233434146563807`*^9, {3.7233434831843004`*^9, 3.7233435093877993`*^9}},
 TextAlignment->Center,
 FontSize->12,
 FontWeight->"Bold"],

Cell["Program code:", "Subsubsection",
 CellDingbat->"\[FilledSmallSquare]",
 CellChangeTimes->{{3.4794189093216*^9, 3.479418932274605*^9}, {
   3.479418981395237*^9, 3.4794189822564754`*^9}, {3.4794192068193808`*^9, 
   3.4794192077607346`*^9}, 3.479420089138093*^9, {3.479420130978256*^9, 
   3.4794201410627565`*^9}, {3.479420560345656*^9, 3.479420578171288*^9}, {
   3.479420683943381*^9, 3.4794206847645617`*^9}, {3.4794210347678413`*^9, 
   3.4794210503903055`*^9}, 3.4803613662913427`*^9, {3.480361406058525*^9, 
   3.48036141974821*^9}, {3.4803614672865667`*^9, 3.480361474717252*^9}, 
   3.4807050675119123`*^9, {3.492804314166378*^9, 3.4928043441496305`*^9}, {
   3.4928044532166224`*^9, 3.492804453513023*^9}, {3.492805162266266*^9, 
   3.492805165713872*^9}, 3.4941724567488375`*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"Int", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"(", 
       RowBox[{"g_", "+", 
        RowBox[{"h_.", "x_"}]}], ")"}], "^", "m_."}], "*", 
     RowBox[{"Gamma", "[", 
      RowBox[{"p_", ",", 
       RowBox[{"f_.", "*", 
        RowBox[{"(", 
         RowBox[{"a_.", "+", 
          RowBox[{"b_.", "*", 
           RowBox[{"Log", "[", 
            RowBox[{"c_.", "*", 
             RowBox[{
              RowBox[{"(", 
               RowBox[{"d_", "+", 
                RowBox[{"e_.", "*", "x_"}]}], ")"}], "^", "n_."}]}], 
            "]"}]}]}], ")"}]}]}], "]"}]}], ",", "x_Symbol"}], "]"}], " ", ":=",
   "\n", "  ", 
  RowBox[{
   RowBox[{
    RowBox[{"1", "/", "e"}], "*", 
    RowBox[{"Subst", "[", 
     RowBox[{
      RowBox[{"Int", "[", 
       RowBox[{
        RowBox[{
         RowBox[{
          RowBox[{"(", 
           RowBox[{"g", "*", 
            RowBox[{"x", "/", "d"}]}], ")"}], "^", "m"}], "*", 
         RowBox[{"Gamma", "[", 
          RowBox[{"p", ",", 
           RowBox[{"f", "*", 
            RowBox[{"(", 
             RowBox[{"a", "+", 
              RowBox[{"b", "*", 
               RowBox[{"Log", "[", 
                RowBox[{"c", "*", 
                 RowBox[{"x", "^", "n"}]}], "]"}]}]}], ")"}]}]}], "]"}]}], 
        ",", "x"}], "]"}], ",", "x", ",", 
      RowBox[{"d", "+", 
       RowBox[{"e", "*", "x"}]}]}], "]"}]}], " ", "/;", "\n", 
   RowBox[{
    RowBox[{"FreeQ", "[", 
     RowBox[{
      RowBox[{"{", 
       RowBox[{
       "a", ",", "b", ",", "c", ",", "d", ",", "e", ",", "f", ",", "g", ",", 
        "h", ",", "m", ",", "n", ",", "p"}], "}"}], ",", "x"}], "]"}], " ", "&&",
     " ", 
    RowBox[{"EqQ", "[", 
     RowBox[{
      RowBox[{
       RowBox[{"e", "*", "g"}], "-", 
       RowBox[{"d", "*", "h"}]}], ",", "0"}], "]"}]}]}]}]], "Code",
 CellChangeTimes->{{3.494097279728842*^9, 3.494097309778884*^9}, {
   3.494099665582182*^9, 3.4940996672021847`*^9}, {3.4940997193022575`*^9, 
   3.494099719662258*^9}, 3.4941724567488375`*^9, 3.494172717389202*^9, 
   3.495161451843186*^9, 3.4951616675135646`*^9, 3.495163619631394*^9, 
   3.495164018025695*^9, 3.495199875426528*^9, 3.495200523437435*^9, 
   3.495200639177597*^9, 3.495200753047757*^9, 3.4952010407881594`*^9, 
   3.4952013363585734`*^9, {3.4952570835699644`*^9, 3.495257087673199*^9}, {
   3.4952590667663965`*^9, 3.495259071698679*^9}, {3.495259155129451*^9, 
   3.4952591595397034`*^9}, {3.4952592284316435`*^9, 3.495259233363926*^9}, {
   3.5705608083492537`*^9, 3.570560819349269*^9}, {3.570560875679348*^9, 
   3.5705608813493557`*^9}, {3.6219001333179817`*^9, 3.62190014570369*^9}, {
   3.621900249252613*^9, 3.621900251408736*^9}, {3.6219002913290195`*^9, 
   3.6219002928911085`*^9}, {3.7164228363883038`*^9, 3.716422849596059*^9}, {
   3.7164229128336763`*^9, 3.7164229352129564`*^9}, 3.7164392531651926`*^9, 
   3.7164392890572453`*^9, 3.716503389679708*^9, 3.7165045499320707`*^9, 
   3.7174656625935264`*^9, {3.717465704865944*^9, 3.717465746281313*^9}, {
   3.7175151029350953`*^9, 3.717515109081506*^9}, {3.717518774473944*^9, 
   3.717518776689148*^9}, {3.7175216087704334`*^9, 3.717521685960848*^9}, {
   3.717521720553827*^9, 3.7175217400349407`*^9}, {3.717521794665066*^9, 
   3.7175218210705757`*^9}, 3.7175222782607255`*^9, {3.7175253204747305`*^9, 
   3.7175253245319624`*^9}, {3.7175260629631987`*^9, 
   3.7175260662003837`*^9}, {3.717526314639593*^9, 3.717526341518131*^9}, {
   3.717526392845066*^9, 3.717526436988591*^9}, {3.717526739113872*^9, 
   3.7175267421670465`*^9}, {3.7176282790592194`*^9, 3.717628331446216*^9}, {
   3.7230837061881156`*^9, 3.723083720287922*^9}, {3.723083756111971*^9, 
   3.723083762583341*^9}, {3.7230839975807824`*^9, 3.723084046227565*^9}, {
   3.72308524831032*^9, 3.723085280867182*^9}, {3.7230890302336335`*^9, 
   3.723089047566625*^9}, 3.7230909790480995`*^9, 3.7230924787078753`*^9, 
   3.7231848887689962`*^9, {3.723184938079817*^9, 3.7231849654193807`*^9}, {
   3.7233434264250536`*^9, 3.723343431848364*^9}, {3.7233434851604133`*^9, 
   3.723343487456545*^9}, {3.7233435191393566`*^9, 3.723343553767337*^9}},
 Background->GrayLevel[0.85]],

Cell["", "Subsubsection",
 CellDingbat->None,
 CellChangeTimes->{3.4796643211106243`*^9, 3.4941724567488375`*^9}]
}, Closed]]
}, Closed]]
},
AutoGeneratedPackage->Automatic,
CellGrouping->Manual,
WindowSize->{1908, 946},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
PrintingCopies->1,
PrintingStartingPageNumber->1,
PrintingPageRange->{1, 32000},
PageHeaders->{{
   Cell[
    TextData[{
      StyleBox[
       CounterBox["Page"], "PageNumber"], " ", "|", " ", 
      StyleBox[
       ValueBox["FileName"], "Header"]}], "Header", 
    CellMargins -> {{0, Inherited}, {Inherited, Inherited}}], None, None}, {
   Cell[
    TextData[{
     "Rules", " ", "for", " ", "integrands", " ", "involving", " ", "gamma", 
      " ", "functions"}], "Header", 
    CellMargins -> {{0, Inherited}, {Inherited, Inherited}}], None, 
   Cell[
    TextData[
     CounterBox["Page", CounterFunction :> Identity]], "Header", 
    CellMargins -> {{Inherited, 0}, {Inherited, Inherited}}]}},
PageFooters->{{None, None, None}, {None, None, None}},
PageHeaderLines->{False, False},
PageFooterLines->{False, False},
PrintingOptions->{"FacingPages"->False,
"FirstPageFace"->Right,
"FirstPageFooter"->False,
"FirstPageHeader"->False,
"Magnification"->1.,
"PaperOrientation"->"Landscape",
"PaperSize"->{792, 612},
"PrintCellBrackets"->False,
"PrintMultipleHorizontalPages"->False,
"PrintRegistrationMarks"->False,
"PrintingMargins"->28.8},
Magnification->1.399999976158142,
FrontEndVersion->"8.0 for Microsoft Windows (64-bit) (October 6, 2011)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[545, 20, 2658, 40, 42, "Subsubtitle"],
Cell[CellGroupData[{
Cell[3228, 64, 3176, 51, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[6429, 119, 4449, 69, 281, "Subsubtitle"],
Cell[10881, 190, 101, 1, 181, "Subsubsection"],
Cell[10985, 193, 836, 26, 221, "Subsubsection"],
Cell[11824, 221, 732, 10, 125, "Subsubsection"],
Cell[12559, 233, 2501, 63, 326, "Subsubtitle"],
Cell[15063, 298, 795, 11, 181, "Subsubsection"],
Cell[15861, 311, 1168, 34, 551, "Code"],
Cell[17032, 347, 113, 2, 181, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[17182, 354, 3450, 56, 281, "Subsubtitle"],
Cell[CellGroupData[{
Cell[20657, 414, 3385, 54, 310, "Subsubtitle"],
Cell[CellGroupData[{
Cell[24067, 472, 3710, 63, 310, "Subsubtitle"],
Cell[CellGroupData[{
Cell[27802, 539, 4783, 73, 310, "Subsubtitle"],
Cell[32588, 614, 2859, 48, 187, "Subsubsection"],
Cell[35450, 664, 996, 14, 125, "Subsubsection"],
Cell[36449, 680, 2663, 59, 604, "Subsubtitle"],
Cell[39115, 741, 795, 11, 181, "Subsubsection"],
Cell[39913, 754, 1565, 38, 551, "Code"],
Cell[41481, 794, 113, 2, 181, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[41631, 801, 5202, 79, 310, "Subsubtitle"],
Cell[46836, 882, 159, 4, 181, "Subsubsection"],
Cell[46998, 888, 564, 14, 193, "Subsubsection"],
Cell[47565, 904, 742, 20, 134, "Subsubsection"],
Cell[48310, 926, 1363, 24, 134, "Subsubsection"],
Cell[49676, 952, 2217, 42, 321, "Subsubtitle"],
Cell[51896, 996, 795, 11, 181, "Subsubsection"],
Cell[52694, 1009, 1476, 33, 551, "Code"],
Cell[54173, 1044, 113, 2, 181, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[54323, 1051, 5494, 88, 310, "Subsubtitle"],
Cell[59820, 1141, 159, 4, 181, "Subsubsection"],
Cell[59982, 1147, 762, 22, 221, "Subsubsection"],
Cell[60747, 1171, 1489, 26, 134, "Subsubsection"],
Cell[62239, 1199, 3139, 74, 338, "Subsubtitle"],
Cell[65381, 1275, 795, 11, 181, "Subsubsection"],
Cell[66179, 1288, 1761, 43, 551, "Code"],
Cell[67943, 1333, 113, 2, 181, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[68093, 1340, 5527, 89, 310, "Subsubtitle"],
Cell[73623, 1431, 159, 4, 181, "Subsubsection"],
Cell[73785, 1437, 807, 23, 214, "Subsubsection"],
Cell[74595, 1462, 1461, 25, 134, "Subsubsection"],
Cell[76059, 1489, 3185, 74, 338, "Subsubtitle"],
Cell[79247, 1565, 795, 11, 181, "Subsubsection"],
Cell[80045, 1578, 1759, 42, 551, "Code"],
Cell[81807, 1622, 113, 2, 181, "Subsubsection"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[81969, 1630, 5126, 82, 310, "Subsubtitle"],
Cell[87098, 1714, 1185, 20, 187, "Subsubsection"],
Cell[88286, 1736, 2262, 50, 326, "Subsubtitle"],
Cell[90551, 1788, 795, 11, 181, "Subsubsection"],
Cell[91349, 1801, 1806, 46, 551, "Code"],
Cell[93158, 1849, 113, 2, 181, "Subsubsection"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[93320, 1857, 4990, 82, 281, "Subsubtitle"],
Cell[98313, 1941, 196, 4, 181, "Subsubsection"],
Cell[98512, 1947, 1261, 26, 221, "Subsubsection"],
Cell[99776, 1975, 1510, 34, 207, "Subsubsection"],
Cell[101289, 2011, 964, 15, 125, "Subsubsection"],
Cell[102256, 2028, 1022, 19, 134, "Subsubsection"],
Cell[103281, 2049, 4885, 133, 1012, "Subsubtitle"],
Cell[108169, 2184, 795, 11, 181, "Subsubsection"],
Cell[108967, 2197, 2028, 61, 687, "Code"],
Cell[110998, 2260, 113, 2, 181, "Subsubsection"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[111160, 2268, 3512, 58, 281, "Subsubtitle"],
Cell[CellGroupData[{
Cell[114697, 2330, 5257, 88, 281, "Subsubtitle"],
Cell[119957, 2420, 165, 4, 181, "Subsubsection"],
Cell[120125, 2426, 1170, 22, 134, "Subsubsection"],
Cell[121298, 2450, 2114, 47, 310, "Subsubtitle"],
Cell[123415, 2499, 795, 11, 181, "Subsubsection"],
Cell[124213, 2512, 2193, 54, 551, "Code"],
Cell[126409, 2568, 113, 2, 181, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[126559, 2575, 5753, 94, 316, "Subsubtitle"],
Cell[132315, 2671, 151, 2, 181, "Subsubsection"],
Cell[132469, 2675, 762, 22, 221, "Subsubsection"],
Cell[133234, 2699, 1595, 28, 134, "Subsubsection"],
Cell[134832, 2729, 3165, 71, 354, "Subsubtitle"],
Cell[138000, 2802, 795, 11, 181, "Subsubsection"],
Cell[138798, 2815, 2537, 69, 551, "Code"],
Cell[141338, 2886, 113, 2, 181, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[141488, 2893, 5697, 97, 281, "Subsubtitle"],
Cell[147188, 2992, 101, 1, 181, "Subsubsection"],
Cell[147292, 2995, 739, 24, 221, "Subsubsection"],
Cell[148034, 3021, 1480, 30, 193, "Subsubsection"],
Cell[149517, 3053, 2702, 69, 349, "Subsubtitle"],
Cell[152222, 3124, 795, 11, 181, "Subsubsection"],
Cell[153020, 3137, 3425, 93, 820, "Code"],
Cell[156448, 3232, 113, 2, 181, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[156598, 3239, 4930, 77, 281, "Subsubtitle"],
Cell[161531, 3318, 992, 14, 181, "Subsubsection"],
Cell[162526, 3334, 1994, 43, 281, "Subsubtitle"],
Cell[164523, 3379, 795, 11, 181, "Subsubsection"],
Cell[165321, 3392, 1903, 45, 551, "Code"],
Cell[167227, 3439, 113, 2, 181, "Subsubsection"]
}, Closed]]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[167401, 3448, 3201, 50, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[170627, 3502, 4531, 69, 550, "Subsubtitle"],
Cell[175161, 3573, 146, 2, 968, "Subsubsection"],
Cell[175310, 3577, 439, 15, 1296, "Subsubsection"],
Cell[175752, 3594, 732, 10, 664, "Subsubsection"],
Cell[176487, 3606, 1453, 30, 1661, "Subsubtitle"],
Cell[177943, 3638, 795, 11, 968, "Subsubsection"],
Cell[178741, 3651, 810, 23, 2958, "Code"],
Cell[179554, 3676, 113, 2, 968, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[179704, 3683, 3388, 56, 550, "Subsubtitle"],
Cell[CellGroupData[{
Cell[183117, 3743, 5082, 84, 550, "Subsubtitle"],
Cell[188202, 3829, 203, 3, 968, "Subsubsection"],
Cell[188408, 3834, 943, 17, 713, "Subsubsection"],
Cell[189354, 3853, 2280, 57, 1721, "Subsubtitle"],
Cell[191637, 3912, 795, 11, 968, "Subsubsection"],
Cell[192435, 3925, 1966, 58, 3687, "Code"],
Cell[194404, 3985, 113, 2, 968, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[194554, 3992, 4956, 77, 550, "Subsubtitle"],
Cell[199513, 4071, 992, 14, 495, "Subsubsection"],
Cell[200508, 4087, 1994, 42, 769, "Subsubtitle"],
Cell[202505, 4131, 795, 11, 495, "Subsubsection"],
Cell[203303, 4144, 1837, 43, 1510, "Code"],
Cell[205143, 4189, 113, 2, 495, "Subsubsection"]
}, Closed]]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[205317, 4198, 3308, 52, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[208650, 4254, 4555, 70, 550, "Subsubtitle"],
Cell[213208, 4326, 146, 2, 968, "Subsubsection"],
Cell[213357, 4330, 476, 16, 1296, "Subsubsection"],
Cell[213836, 4348, 732, 10, 664, "Subsubsection"],
Cell[214571, 4360, 1538, 32, 1661, "Subsubtitle"],
Cell[216112, 4394, 795, 11, 968, "Subsubsection"],
Cell[216910, 4407, 879, 24, 2958, "Code"],
Cell[217792, 4433, 113, 2, 968, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[217942, 4440, 3470, 58, 550, "Subsubtitle"],
Cell[CellGroupData[{
Cell[221437, 4502, 5041, 83, 550, "Subsubtitle"],
Cell[226481, 4587, 101, 1, 968, "Subsubsection"],
Cell[226585, 4590, 987, 18, 713, "Subsubsection"],
Cell[227575, 4610, 2350, 58, 1721, "Subsubtitle"],
Cell[229928, 4670, 795, 11, 968, "Subsubsection"],
Cell[230726, 4683, 2110, 60, 2958, "Code"],
Cell[232839, 4745, 113, 2, 968, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[232989, 4752, 4958, 82, 550, "Subsubtitle"],
Cell[237950, 4836, 163, 2, 968, "Subsubsection"],
Cell[238116, 4840, 903, 17, 713, "Subsubsection"],
Cell[239022, 4859, 2461, 63, 1873, "Subsubtitle"],
Cell[241486, 4924, 795, 11, 968, "Subsubsection"],
Cell[242284, 4937, 2279, 68, 3687, "Code"],
Cell[244566, 5007, 113, 2, 968, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[244716, 5014, 4962, 78, 550, "Subsubtitle"],
Cell[249681, 5094, 992, 14, 495, "Subsubsection"],
Cell[250676, 5110, 2054, 44, 769, "Subsubtitle"],
Cell[252733, 5156, 795, 11, 495, "Subsubsection"],
Cell[253531, 5169, 1911, 45, 1510, "Code"],
Cell[255445, 5216, 113, 2, 495, "Subsubsection"]
}, Closed]]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[255619, 5225, 4655, 73, 53, "Subsubtitle"],
Cell[260277, 5300, 146, 2, 968, "Subsubsection"],
Cell[260426, 5304, 618, 21, 1210, "Subsubsection"],
Cell[261047, 5327, 837, 12, 664, "Subsubsection"],
Cell[261887, 5341, 1707, 36, 1749, "Subsubtitle"],
Cell[263597, 5379, 795, 11, 968, "Subsubsection"],
Cell[264395, 5392, 1075, 30, 2958, "Code"],
Cell[265473, 5424, 113, 2, 968, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[265623, 5431, 5016, 84, 53, "Subsubtitle"],
Cell[270642, 5517, 146, 2, 968, "Subsubsection"],
Cell[270791, 5521, 711, 22, 1210, "Subsubsection"],
Cell[271505, 5545, 969, 18, 713, "Subsubsection"],
Cell[272477, 5565, 1897, 42, 1749, "Subsubtitle"],
Cell[274377, 5609, 795, 11, 968, "Subsubsection"],
Cell[275175, 5622, 1418, 42, 2958, "Code"],
Cell[276596, 5666, 113, 2, 968, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[276746, 5673, 3764, 61, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[280535, 5738, 4891, 77, 53, "Subsubtitle"],
Cell[285429, 5817, 101, 1, 48, "Subsubsection"],
Cell[285533, 5820, 1658, 46, 69, "Subsubsection"],
Cell[287194, 5868, 880, 12, 34, "Subsubsection"],
Cell[288077, 5882, 3328, 74, 104, "Subsubtitle"],
Cell[291408, 5958, 795, 11, 48, "Subsubsection"],
Cell[292206, 5971, 2872, 73, 145, "Code"],
Cell[295081, 6046, 113, 2, 48, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[295231, 6053, 4976, 79, 65, "Subsubtitle"],
Cell[300210, 6134, 161, 2, 48, "Subsubsection"],
Cell[300374, 6138, 1398, 33, 60, "Subsubsection"],
Cell[301775, 6173, 879, 12, 34, "Subsubsection"],
Cell[302657, 6187, 2441, 51, 90, "Subsubtitle"],
Cell[305101, 6240, 795, 11, 48, "Subsubsection"],
Cell[305899, 6253, 1971, 46, 145, "Code"],
Cell[307873, 6301, 113, 2, 48, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[308023, 6308, 5294, 89, 53, "Subsubtitle"],
Cell[313320, 6399, 101, 1, 35, "Subsubsection"],
Cell[313424, 6402, 1757, 48, 50, "Subsubsection"],
Cell[315184, 6452, 941, 19, 26, "Subsubsection"],
Cell[316128, 6473, 4187, 101, 74, "Subsubtitle"],
Cell[320318, 6576, 795, 11, 35, "Subsubsection"],
Cell[321116, 6589, 4028, 109, 129, "Code"],
Cell[325147, 6700, 113, 2, 35, "Subsubsection"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[325309, 6708, 3875, 65, 53, "Subsubtitle"],
Cell[CellGroupData[{
Cell[329209, 6777, 6385, 100, 53, "Subsubtitle"],
Cell[335597, 6879, 478, 7, 67, "Subsubsection"],
Cell[336078, 6888, 1923, 27, 47, "Subsubsection"],
Cell[338004, 6917, 3747, 73, 114, "Subsubtitle"],
Cell[341754, 6992, 795, 11, 67, "Subsubsection"],
Cell[342552, 7005, 3655, 74, 202, "Code"],
Cell[346210, 7081, 113, 2, 67, "Subsubsection"]
}, Closed]],
Cell[CellGroupData[{
Cell[346360, 7088, 6928, 116, 53, "Subsubtitle"],
Cell[353291, 7206, 478, 7, 35, "Subsubsection"],
Cell[353772, 7215, 2284, 56, 39, "Subsubsection"],
Cell[356059, 7273, 2206, 37, 26, "Subsubsection"],
Cell[358268, 7312, 4093, 82, 59, "Subsubtitle"],
Cell[362364, 7396, 795, 11, 35, "Subsubsection"],
Cell[363162, 7409, 4174, 91, 104, "Code"],
Cell[367339, 7502, 113, 2, 35, "Subsubsection"]
}, Closed]]
}, Closed]]
}
]
*)

(* End of internal cache information *)