Computability of general integrals and integral transforms

Oleg Marichev    (Wolfram Research Inc.)

The majority of integrals (and classical integral transforms) that have been evaluated in numerous publications can be generalized to integrals and transforms 
that include the MeijerG function, either in integrands or in kernels of integral transforms. In the case of indefinite integrals one can also consider integrals 
with factors of the form Rj[x]^aj, where Rj[x] are arbitrary rational functions.
          In the talk we classify such general integrals and transforms and demonstrate how to evaluate them in closed form.

Blogs

New Derivatives of the Bessel Functions Have Been Discovered with the Help of the Wolfram Language

others

Wolfram Functions Site (307000+ formulas)

Wolfram Functions Site

Developers

My first book of 1978 (by Amazon & from Editor)

From user

A Great Source for Those Who Want to Solve Integral transforms and equations

This book is a great introduction to the Mellin Transform and Hypergeometric functions.If you have ever experienced impatience with conventional tables of integrals or have obtained a transform not listed in the tables, this is a great source for you.You will have to study it and apply its lessons to the problems you encounter, but I much prefer this approach to dryly sifting through a table of integrals.There is nothing to be learned by this activity.You will also learn a lot about special functions For instance that almost all of them them belong to the general hypergeometric family.Instead of letting investigations lapse because you can' t find or solve an integral transform or equation, you can proceed with an approach of your choosing.Marichev also includes a great series of tables that list many relatively new results.He also discusses many examples that show you how to proceed on your own.I have owned a copy since 1983 and I use it constantly.

from Editor F.D. Gakhov

This reference book by O. T. Marichev does not have a parallel in the Russian or foreign literature, the principal distinction in contrast to other known reference works on integrals being its dynamic character. Each integral given in it can be a source for the derivation of a large number of other integrals; these are obtained from the given one by using fairly straightforward transformations, and the requisite rules are explained in sufficient detail in the text.

Front pages from book

Translated by L. W. LONGDON Royal Military College of Science, Shrivenham

ELLIS HORWOOD LIMITED Publishers • Chichester
Halsted Press: a division of JOHN WILEY & SONS
New York • Brisbane • Chichester ■ Toronto

First published in 1983 by ELLIS HORWOOD LIMITED
Market Cross House, Cooper Street, Chichester, West Sussex, P019 1EB, England
The publisher’s colophon is reproduced from James Gillison’s drawing of the ancient Market Cross, Chichester.
Distributors:
Australia, New Zealand, South-east Asia: Jacaranda-Wiley Ltd., Jacaranda Press,
JOHN WILEY & SONS INC.,
G.P.O. Box 859, Brisbane, Queensland 40001, Australia Canada:
JOHN WILEY & SONS CANADA LIMITED 22 Worcester Road, Rexdale, Ontario, Canada.
Europe, Africa:
JOHN WILEY & SONS LIMITED
Baffins Lane, Chichester, West Sussex, England.
North and South America and the rest of the world: Halsted Press: a division of JOHN WILEY & SONS
605 Third Avenue, New York, N.Y. 10016, U.S.A.
© 1983 O. I. Marichev/Ellis Horwood Ltd.
British Library Caloguing in Publication Data
Marichev, O. I.
Handbook of integral transforms of higher transcendental functions. - (Ellis Horwood series in mathematics and its applications)
1. Integral transforms
I. Title II. Metod vychisleniya integralov ot spetsial ‘nykh funktsii. English 515’.723 QA432
Library of Congress Card No. 82-15849
ISBN 0-85312-538-7 (Ellis Horwood Limited) ISBN 0-470-27 364-X (Halsted Press)
Printed in Great Britain by Butler & Tanner, Frome, Somerset
English edition © Ellis Horwood Limited 1982.
Originally published in Russian as Metod vychisleniya integralov ot spetsial’nykh funktsii (Teoriya i tablitsy formul) by the Minsk Science and Engineering Publishing House 1978.
First published in English by Ellis Horwood Limited 1982.
COPYRIGHT NOTICE -
All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the permission of Ellis Horwood Limited, Market Cross House, Cooper Street, Chichester, West Sussex, England.

18 books (picture)

7 PhD students

Vu Kim Tuan

Yakubovich S.B.

Adamchik V.S.

Nguyen Thanh Hai

MeijerGReduce

Function  MeijerG

MeijerG[{{(Subscript[a, 1]),…,(Subscript[a, n])},{(Subscript[a, n+1]),…,(Subscript[a, p])}},{{(Subscript[b, 1]),…,(Subscript[b, m])},{(Subscript[b, m+1]),…,(Subscript[b, q])}},z]==(1/(2 π I))ContourIntegrate[((((∏_(k=1)^m)Gamma[(Subscript[b, k])+s]) (∏_(k=1)^n)Gamma[1-(Subscript[a, k])-s] )/(((∏_(k=n+1)^p)Gamma[(Subscript[a, k])+s]) (∏_(k=m+1)^q)Gamma[1-(Subscript[b, k])-s]))(z^(-s)),{s,ℒ}]/;m∈Integers∧m≥0∧ n∈Integers∧n≥0∧p∈Integers∧p≥0∧q∈Integers∧q≥0∧m≤q∧n≤p

The infinite contour of integration ℒ separates the poles of Γ(1-(Subscript[a, k])-s)at s=1-(Subscript[a, k])+j, j∈ℕ from the poles of Γ((Subscript[b, i])+ s) at s=-(Subscript[b, i])-l, l∈ℕ. Such a contour always exists in the cases  (Subscript[a, k])-(Subscript[b, i])-1∉ℕ.

There are three possibilities for the contour ℒ :

(i) ℒ runs from γ-i ∞  to γ+i  ∞ (where Im(γ)=0) so that all poles of Γ((Subscript[b, i])+ s), i=1,...,m, are to the left, and all the poles of Γ(1-(Subscript[a, i])- s), i=1,...,n, to the right, of ℒ. 
This contour can be a straight  line (γ-i ∞, γ+i ∞) if  Re((Subscript[b, i])-(Subscript[a, k]))>-1 (then  -Re((Subscript[b, i]))<γ<1-Re((Subscript[a, k]))). (In this case the integral converges if p+q<2(m+n) ,  |Arg(z)|<(m +n-((p+q)/2)) π. If m +n-((p+q)/2)=0, then z must be real and positive and additional condition  (q-p)γ+Re(μ)<0,μ==(∑_(l=1)^q)(Subscript[b, l])-(∑_(k=1)^p)(Subscript[a, k])+((p-q)/2)+1, should be added.)

(ii) ℒ is a left loop, starting and ending at -∞ and encircling all  poles of Γ((Subscript[b, i])+ s), i=1,...,m, once in the positive direction, but none of the poles of  Γ(1-(Subscript[a, i])- s), i=1,...,n. 
(In this case the integral converges if q≥1 and either q>p or q=p and |z|<1 or q=p and |z|==1and  m +n-((p+q)/2)≥0 and  Re(μ)<0.)

(iii) ℒ is a right loop, starting and ending at +∞ and encircling all  poles of Γ(1-(Subscript[a, i])-s), i=1,...,n, once in the negative direction, but none of the poles of Γ((Subscript[b, i])+s), i=1,...,m. 
(In this case the integral converges if p≥1 and either p>q or p=q and \[LeftBracketingBar]z\[RightBracketingBar]>1 or q=p and \[LeftBracketingBar]z\[RightBracketingBar]==1and  m +n-((p+q)/2)≥0 and  Re(μ)<0.)

Special notations for this file:

Examples of MeijerGReduce

??MeijerGReduce

MeijerGReduce[Sin[x],x]

MeijerGReduce[(((1+x))^β),x]

MeijerGForm2 (program) (New structure)

li5162 (edited li5310)

Conversion li5162 ->MeijerM  (==Inactive[MeijerG])

{#,MeijerGForm2[#,z]}&/@Table[li5162[[kk]],{kk,RandomInteger[{1,5162},6]}]//TableForm

{#,MeijerGForm2[#,z]}&/@li5162//Timing

{11.8872762`,{{1,MeijerM[{{},{}},{{0},{}},z]+MeijerM[{{1},{}},{{1},{0}},z]},{(E^(-1/z)),MeijerM[{{},{}},{{0},{}},(1/z)]},{(E^(-(z^(1/4)))),MeijerM[{{},{}},{{0},{}},(z^(1/4))]},{(E^(z^(1/4))),4 (Sqrt[2]) (π^(5/2)) MeijerM[{{},{(1/8),(3/8),(5/8),(7/8)}},{{0,(1/4),(1/2),(3/4)},{(1/8),(3/8),(5/8),(7/8)}},(z/256)]},…}}

Select[%[[2]],FreeQ[#,MeijerM]&]

Length[li5162]

Integral from one MeijerG

Integral of one MeijerG && Gamma

(∫_0^∞)(t^(α-1))MeijerG[{{(Subscript[a, 1]),…,(Subscript[a, n])},{(Subscript[a, n+1]),…,(Subscript[a, p])}},{{(Subscript[b, 1]),…,(Subscript[b, m])},{(Subscript[b, m+1]),…,(Subscript[b, q])}},t z]\[DifferentialD]t==(z^(-α)) (((∏_(k=1)^m)Gamma[(Subscript[b, k])+α](∏_(k=1)^n)Gamma[1-(Subscript[a, k])-α])/((∏_(k=n+1)^p)Gamma[(Subscript[a, k])+α] (∏_(k=m+1)^q)Gamma[1-(Subscript[b, k])-α]))/;(SuperStar[c])==m+n-((p+q)/2)∧(m^2)+(n^2)>0∧z≠0∧(((SuperStar[c])>0∧Abs[Arg[z]]<(SuperStar[c])π∧-Min[Re[(Subscript[b, 1])],…,Re[(Subscript[b, m])]]<Re[α]<1-Max[Re[(Subscript[a, 1])],…,Re[(Subscript[a, n])]])∨(p≠q∧(SuperStar[c])≥0∧Abs[Arg[z]]==(SuperStar[c])π∧-Min[Re[(Subscript[b, 1])],…,Re[(Subscript[b, m])]]<Re[α]<1-Max[Re[(Subscript[a, 1])],…,Re[(Subscript[a, n])]]∧Re[μ+(q-p)α]<(3/2)∧μ==(∑_(j=1)^q)(Subscript[b, j])-(∑_(j=1)^p)(Subscript[a, j])+((p-q)/2)+1)∨(p==q∧(SuperStar[c])==0∧z>0∧-Min[Re[(Subscript[b, 1])],…,Re[(Subscript[b, m])]]<Re[α]<1-Max[Re[(Subscript[a, 1])],…,Re[(Subscript[a, n])]]∧Re[(∑_(j=1)^q)((Subscript[b, j])-(Subscript[a, j]))]<0))

IntegralOfOneMeijerG  (∫_0^∞)(t^(α+q0)) MeijerM[{} {},z (t^q1),r]dt  (program)

Evaluation integrals from li5162

Clear[set];set=RandomInteger[{1,5162},6];
Table[{Integral[(t^(α-1))li5162[[kk]]/.{z->z  (t^r)},{t, 0,Infinity}],IntegralOfOneMeijerG[α-1,MeijerGForm0[li5162[[kk]],z (t^r),z]/.{(((uu_.(t^(ss_.)) vv_.))^(rr_)):>(((uu vv))^rr)(t^(ss rr))},t, 0,Infinity]},{kk,set}]//TableForm

Table[{Integral[(t^(α-1))li5162[[kk]]/.{z->z  (t^r)},{t, 0,Infinity}],IntegralOfOneMeijerG[α-1,MeijerGForm0[li5162[[kk]],z (t^r),z]/.{(((uu_.(t^(ss_.)) vv_.))^(rr_)):>(((uu vv))^rr)(t^(ss rr))},t, 0,Infinity]},{kk,1,5162}]//Timing

Select[%[[2]], FreeQ[#,IntegralOfOneMeijerG]&]//Length

3098/5162//N

{Integral[(t^(α-1))#/.{z->z  (t^r)},{t, 0,Infinity}],IntegralOfOneMeijerG[α-1,MeijerGForm0[#,z (t^r),z]/.{(((uu_.(t^(ss_.)) vv_.))^(rr_)):>(((uu vv))^rr)(t^(ss rr))},t, 0,Infinity]}&/@{CosIntegral[(Sqrt[z])],SinIntegral[(Sqrt[z])]-(π/2),(((SinIntegral[z]-(π/2)))^2)+((CosIntegral[z])^2)}//TableForm

IntegralOfTwoMeijerG

Integral of two MeijerG && HypergeometricPFQ

(∫_0^∞)(τ^(α-1))MeijerG[{{(Subscript[c, 1]),…,(Subscript[c, t])},{(Subscript[c, t+1]),…,(Subscript[c, u])}},{{(Subscript[d, 1]),…,(Subscript[d, s])},{(Subscript[d, s+1]),…,(Subscript[d, v])}},τ w]MeijerG[{{(Subscript[a, 1]),…,(Subscript[a, n])},{(Subscript[a, n+1]),…,(Subscript[a, p])}},{{(Subscript[b, 1]),…,(Subscript[b, m])},{(Subscript[b, m+1]),…,(Subscript[b, q])}},τ z]\[DifferentialD]τ==(w^(-α))MeijerG[{{(Subscript[a, 1]),…,(Subscript[a, n]),1-α-(Subscript[d, 1]),…,1-α-(Subscript[d, s])},{1-α-(Subscript[d, s+1]),…,1-α-(Subscript[d, v]),(Subscript[a, n+1]),…,(Subscript[a, p])}},{{(Subscript[b, 1]),…,(Subscript[b, m]),1-α-(Subscript[c, 1]),…,1-α-(Subscript[c, t])},{1-α-(Subscript[c, t+1]),…,1-α-(Subscript[c, u]),(Subscript[b, m+1]),…,(Subscript[b, q])}},(z/w)]/;(ℂℂ/;k==l==1)

AsymptoticG101 with conditions Final NEW4    RUN 4  (program with 43 groups of conditions)

IntegralOfTwoMeijerG (=Int3)  (program)

Evaluation integrals from li5162

IntegralOfTwoMeijerG[s-1,#,t]&/@{{(((t+1))^α),Gamma[ν,t]},{Sin[t],Gamma[ν,t]},{Exp[-t],BesselK[ν,t]}}

IntegralOfTwoMeijerG[s-1,#,t]&/@{{Exp[-t],(((t+1))^α)},{Exp[-t],Gamma[ν,t]},{Exp[-t],BesselK[ν,t]}}/.z1->p

With[{MeijerM=MeijerG,Integral=NIntegrate,ν=Random[Complex],p=Random[Complex],z2=Random[Complex],s=Random[Complex]},{Integral[(E^(-p t)) (t^(-1+s)) BesselK[ν,t z2],{t,0,∞}],(((2^(-2+s)) (p^(-s)) MeijerM[{{1-(ν/2),((2+ν)/2)},{}},{{((1+s)/2),(s/2)},{}},((p^2)/(z2^2)),1])/(Sqrt[π]))}]

LaplaceTransform from li5162 Integral[(e^(-p t)) (t^(s-1)) li5162[t],{t,o,∞}]

function5162={Exp[-t],#}&/@(li5162/.{z->(t^r)})

(IntegralOfTwoMeijerG[s-1,#,t]&/@function5162)/.z1->p

Select[%,FreeQ[#,Integrate2]&]//Length

4335/5162//N

HankelTransform from li5162  Integral[t BesselJ[ν,x t] (t^s) li5162[t],{t,0,∞}]

Function5162={BesselJ[ν1,(Sqrt[t])],#}&/@(li5162/.{z->t})

Timing[(IntegralOfTwoMeijerG[s-1,#,t]&/@Function5162)/.z1->(x^2)]

{184.9859858`,{{Integral[(t^(-1+s)) BesselJ[ν1,(Sqrt[t (x^2)])],{t,0,∞}],Integrate2[-1+s,MeijerM[{{},{}},{{((ν1)/2)},{-((ν1)/2)}},((Sqrt[t (x^2)])/2),(1/2)] (MeijerM[{{},{}},{{0},{}},t z2,1]+MeijerM[{{1},{}},{{1},{0}},t z2,1]),{t,0,∞},False,True]},{Integral[(E^(-(1/(t z2)))) (t^(-1+s)) BesselJ[ν1,(Sqrt[t (x^2)])],{t,0,∞}],ConditionalExpression[(4^s) ((((x^2)))^(-s)) MeijerM[{{1,1-s-((ν1)/2)},{1-s+((ν1)/2)}},{{},{}},((4 z2)/(x^2))],$Failed]},{Integral[(E^(-(((t z2))^(1/4)))) (t^(-1+s)) BesselJ[ν1,(Sqrt[t (x^2)])],{t,0,∞}],ConditionalExpression[(((2^(-(1/2)+8 s)) (z2^(-s)) MeijerM[{{(1/4)-s,(1/2)-s,(3/4)-s,1-s},{}},{{((ν1)/2)},{-((ν1)/2)}},((64 (x^2))/z2)])/(π^(3/2))),$Failed]},{Integral[(E^(((t z2))^(1/4))) (t^(-1+s)) BesselJ[ν1,(Sqrt[t (x^2)])],{t,0,∞}],(2^((5/2)+8 s)) (π^(5/2)) (z2^(-s)) MeijerM[{{1-((ν1)/2)},{(1/8)+s,(3/8)+s,(5/8)+s,(7/8)+s,1+((ν1)/2)}},{{s,(1/4)+s,(1/2)+s,(3/4)+s},{(1/8)+s,(3/8)+s,(5/8)+s,(7/8)+s}},(z2/(64 (x^2))),1]}}…}

Select[%[[2]],(FreeQ[#,Integrate1]&&FreeQ[#,Integrate2])&]//Length

4311/5162//N

With[{MeijerM=MeijerG,Integral=NIntegrate,ν1=Random[Complex],a=Random[Complex],μ=Random[Complex],ν=Random[Complex],z2=Random[Complex],x=Random[Complex],s=Random[Complex]},{Integral[(t^(-1+s)) (-BesselI[-μ,t z2] BesselI[-ν,t z2]+BesselI[μ,t z2] BesselI[ν,t z2]) BesselJ[ν1,(Sqrt[t (x^2)])],{t,0,1000}],ConditionalExpression1[-(1/(π^(3/2)))(2^(-1+4 s)) ((((x^2)))^(-s)) MeijerM[{{(1/2) (2-μ-ν),(1/2) (2+μ+ν)},{(1/2) (2-μ+ν),(1/2) (2+μ-ν)}},{{1,(1/2),(1/4) (2+2 s+ν1),(1/4) (2 s+ν1)},{(1/4) (2+2 s-ν1),(1/4) (2 s-ν1)}},((x^4)/(256 (z2^2))),1] Sin[π (μ+ν)],μ+ν∉(Integers)]}]

Integral[(e^(-t z1)) (t^(-1+s)) GammaRegularized[a,0,(t^2) (z2^2)],{t,0,∞}] (Frozen)

{Exp[-t],#}&/@(Table[li5162[[kk]],{kk,RandomInteger[{1,5162},3]}]/.{z->(t^2)})

IntegralOfTwoMeijerG[s-1,#,t]&/@%

With[{MeijerM=MeijerG,Integral=NIntegrate,a=Random[Complex],b=Random[Complex],ν=Random[Complex],z1=Random[Complex],z2=Random[Complex],s=Random[Complex]},{Integral[(E^(-t z1)) (t^(-1+s)) GammaRegularized[a,0,(t^2) (z2^2)],{t,0,∞}],(((2^(-1+s)) (z1^(-s)) MeijerM[{{1-a},{1}},{{0,((1+s)/2),(s/2)},{}},((z1^2)/(4 (z2^2))),1])/((Sqrt[π]) Gamma[a]))}]

IntegralOfThreeMeijerG also

Integral of three MeijerG && Kampé de Fériet function (see blog)

New Derivatives of the Bessel Functions Have Been Discovered with the Help of the Wolfram Language

Below we demonstrate integral  from products of three Meijer G-functions:

(∫_0^∞)(t^(α-1)) (Subscript[G, p,q]^(m,n))(z t|({{(Subscript[a, 1]),}, {(Subscript[b, 1]),}})({{…,, (Subscript[a, p])}, {…,, (Subscript[b, q])}}))(Subscript[G, (Subscript[p, 1]),(Subscript[q, 1])]^((Subscript[m, 1]),(Subscript[n, 1])))(x t|({{(Subscript[a, 11]),}, {(Subscript[b, 11]),}})({{…,, (Subscript[a, 1(Subscript[p, 1])])}, {…,, (Subscript[b, 1(Subscript[q, 1])])}}))(Subscript[G, (Subscript[p, 2]),(Subscript[q, 2])]^((Subscript[m, 2]),(Subscript[n, 2])))(y t|({{(Subscript[a, 21]),}, {(Subscript[b, 21]),}})({{…,, (Subscript[a, 2(Subscript[p, 2])])}, {…,, (Subscript[b, 2(Subscript[q, 2])])}}))dt==(z^(-α))(Subscript[G, q,p:(Subscript[p, 1]),(Subscript[q, 1]):(Subscript[p, 2]),(Subscript[q, 2])]^(n,m:(Subscript[m, 1]),(Subscript[n, 1]):(Subscript[m, 2]),(Subscript[n, 2])))(({{1-α-(Subscript[b, 1]),}, {1-α-(Subscript[a, 1]),}})({{…,, 1-α-(Subscript[b, q])}, {…,, 1-α-(Subscript[a, p])}})|({{(Subscript[a, 11]),}, {(Subscript[b, 11]),}})({{…,, (Subscript[a, 1(Subscript[p, 1])])}, {…,, (Subscript[b, 1(Subscript[q, 1])])}})|({{(Subscript[a, 21]),}, {(Subscript[b, 21]),}})({{…,, (Subscript[a, 2(Subscript[p, 2])])}, {…,, (Subscript[b, 2(Subscript[q, 2])])}})|(x/z),(y/z))

The right side of this formula includes a Meijer-G function of two variables that generically can be represented, in the non-logarithmic cases, as a finite sum of Kampé de Fériet hypergeometric functions with some coefficients, by analogy with the formulas http://functions.wolfram.com/HypergeometricFunctions/MeijerG/06/01/03/01/0004/ and http://functions.wolfram.com/HypergeometricFunctions/MeijerG/06/01/05/01/0004/.

Kampé de Fériet hypergeometric function

Here the Kampé de Fériet hypergeometric function is defined by the double series (see https://en.wikipedia.org/wiki/Kamp% C3 % A9_de _F % C3 % A9riet_function and http://functions.wolfram.com/Notations/5/)

(Subscript[F, P,Q,S]^(A,B,C))(({{(Subscript[a, 1]),…,(Subscript[a, A]);(Subscript[b, 1]),…,(Subscript[b, B]);(Subscript[c, 1]),…,(Subscript[c, C]);}, {(Subscript[p, 1]),…,(Subscript[p, P]);(Subscript[q, 1]),…,(Subscript[q, Q]);(Subscript[s, 1]),…,(Subscript[s, S]);}})z,w)==(∑_(m=0)^∞)(∑_(n=0)^∞)(((∏_(j=1)^A)(Subscript[((Subscript[a, j])), m+n]) (∏_(j=1)^B)(Subscript[((Subscript[b, j])), m]) (∏_(j=1)^C)(Subscript[((Subscript[c, j])), n]) (z^m)(w^n))/(m! n! (∏_(j=1)^P)(Subscript[((Subscript[p, j])), m+n]) (∏_(j=1)^Q)(Subscript[((Subscript[q, j])), m]) (∏_(j=1)^S)(Subscript[((Subscript[s, j])), n])))/;(A+B≤P+Q∧A+C≤P+S∧(Abs[z])<∞∧(Abs[w])<∞)∨(A+B==P+Q+1∧A+C<P+S+1∧(Abs[z])<1∧(Abs[w])<∞)∨(A+B<P+Q+1∧A+C==P+S+1∧(Abs[z])<∞∧(Abs[w])<1)∨(A+B==P+Q+1∧A+C==P+S+1∧(Abs[z])<1∧(Abs[w])<1∧A≤P)∨(A+B==P+Q+1∧A+C==P+S+1∧((Abs[w])^(1/(A-P)))+((Abs[z])^(1/(A-P)))<1∧A>P)

The Kampé de Fériet function can be considered as a generalization of the hypergeometric function to two variables:

KampeDeFeriet[{{(Subscript[a, 1]),…,(Subscript[a, A])},{(Subscript[b, 1]),…,(Subscript[b, B])},{(Subscript[c, 1]),…,(Subscript[c, C])}},{{(Subscript[p, 1]),…,(Subscript[p, P])},{(Subscript[q, 1]),…,(Subscript[q, Q])},{(Subscript[s, 1]),…,(Subscript[s, S])}},z,w]==HypergeometricPFQ[{{(Subscript[a, 1]),…,(Subscript[a, A])},{(Subscript[b, 1]),…,(Subscript[b, B])},{(Subscript[c, 1]),…,(Subscript[c, C])}},{{(Subscript[p, 1]),…,(Subscript[p, P])},{(Subscript[q, 1]),…,(Subscript[q, Q])},{(Subscript[s, 1]),…,(Subscript[s, S])}},z,w]

A corresponding regularized version of the function can also be defined by replacing the product of Pochhammer symbols (∏_(j=1)^P)(Pochhammer[(Subscript[p, j]), m+n]) (∏_(j=1)^Q)(Pochhammer[(Subscript[q, j]), m]) (∏_(j=1)^S)(Pochhammer[(Subscript[s, j]), n]) in the denominator with (∏_(j=1)^P)(Gamma[(m+n)+(Subscript[p, j])]) (∏_(j=1)^Q)(Gamma[m+(Subscript[q, j])]) (∏_(j=1)^S)(Gamma[n+(Subscript[s, j])]).

The Kampé de Fériet function can be used to obtain a representation of the derivatives of the Bessel function J with respect to its parameter; i.e.,

(Subscript[J, ν]^((1,0)))(z)==(((2^(-ν-2)) (z^(ν+2)))/((ν+1) (Gamma[ν+2])))(Subscript[F, 1,2,0]^(0,2,0))(({{;ν+1,ν+1;;}, {2;ν+2,ν+2;;}})((z^2)/4),-((z^2)/4))+(log((z/2))-(PolyGamma[ν+1]))(BesselJ[ν, z])

Program with 204 groups of conditions exists but not tested well

General Indefinite Integrals

Simple integrals

∫(x^(μ-1)) (((a (x^ν)+b))^α)\[DifferentialD]x

∫(x^(μ-1))(((a (x^(2ν))+b (x^ν)+c))^α)\[DifferentialD]x

∫ (((a+p x))^α) (((b+q x))^β) (((c+r x))^γ)\[DifferentialD]x

∫ (((a (x^3)+b (x^2)+c x+d))^γ)\[DifferentialD]x

D[%,x]//FullSimplify

Integral with 13 parameters

Integral with 19 parameters

Doable general trigonometric integrals

General Integral Transforms

How to find “molicular theory” of general integral transforms

Let us open largest wight book from the pictures of 18 books above

Its authors are Samko S.G., Kilbas A.A., Marichev O.I.
 The title is  “Integrals and derivatives of fractional order and some of their applications”, Gordon and Breach Science Publishers, c1993, 976 p. More comments see at Mathematical Reviews

Chapter 7 of this book shows that majority classical integral transforms can be expressed as particular cases of G- and W- transforms with MeijerG function at the kernels. All those transforms have “molecular structure” and can be decomposed and solved through compositions of only two “atomic” transforms with names direct and inverse Laplace transforms and multiplicators of type x^α  between them. It can be realized at the new class of functions (generalized famous (Subscript[L, p]))  through concrete formulas.