a method to compute integral

I am trying to calculate the following integral. But, Mathematica does not compute it

qqq = (E^(-((a x + b x)/(a b x^2)) + s/(a x + b x + y)) ((a x)/4 + (b x)/4 + y/4)^(-D/2)Log[y])/(a x + b x + y)^9;

qqq = Integrate[qqq, {x, 0, Infinity},GenerateConditions -> False] // Expand
qqq = Integrate[qqq, {y, 0, Infinity},GenerateConditions -> False] // Expand


I tried to use Jacobian (change of variables in double Integral), but I can't find appropriate parameters. Is there any special method to help me?

• it is not a good idea to use (-D/2). Better use d. But your integral does not seem to be solvable analytically. Dec 7, 2021 at 12:13

qqq = (E^(-((a x + b x)/(a b x^2)) + s/(a x + b x + y)) ((a x)/4 + (b x)/4 +
y/4)^(-d/2) Log[y])/(a x + b x + y)^9;


Looking at the much simpler case for when b == -a

qqq2 = qqq /. b -> -a

(* 2^d E^(s/y) y^(-9 - d/2) Log[y] *)


This is a function only of y

Assuming[s < 0 && d > -16,
Integrate[qqq2, {y, 0, Infinity}]]

(* 2^d (-s)^(-8 - d/2) Gamma[8 + d/2] (Log[-s] - PolyGamma[0, 8 + d/2]) *)

• dear @Bob Hanlon, the problem is that b is not equal to -a
– asal
Dec 8, 2021 at 8:48