# Help Evaluating Infinite Definite Integral

I am reading a publication and trying to follow all of the derivations, but I am having trouble evaluating the following integral which they say has a solution!

Integrate[(1/2 (phiM + x + y)^2 +
phiM^3/24 (3 phiM^2 + 12 phiM*x + 12 x^2)) (((phiM + x + y)^2 +
phiM^2 (3/4 phiM^2 + x^2 + 2 x*phiM + y*phiM + 2 x*y))/((phiM +
x + y)^2 + (1/4 phiM^2) (phiM + 2 x)^2)^3 -
1/((phiM + x +
y)^2 + (1/12 phiM^2) (phiM^2 + 4 x*phiM + 4 y*phiM +
12 x*y))^2), {y, 0, Infinity}]


Sorry I know it looks like a mess, but the integral is integrated over y from 0 to Infinity and x, phiM are both constants of integration. Any help would be greatly appreciated!

Below is the integral in a more simpler view:

• Adding the equation in the math form might help. Commented Nov 3, 2016 at 20:27
• @anderstood ok done! Commented Nov 3, 2016 at 20:40
• Any assumptions on x, phiM? Real? Positive? Commented Nov 3, 2016 at 21:06
• @MichaelE2 I've tried those assumptions in the integration but it did not help :( Commented Nov 3, 2016 at 21:13
• Who is this they? Commented Nov 3, 2016 at 21:36

I successfully integrated it with the assumptions as follows:

    int=Integrate[(1/2 (phiM + x + y)^2 +
phiM^3/24 (3 phiM^2 + 12 phiM*x + 12 x^2)) (((phiM + x + y)^2 +
phiM^2 (3/4 phiM^2 + x^2 + 2 x*phiM + y*phiM +
2 x*y))/((phiM + x +
y)^2 + (1/4 phiM^2) (phiM + 2 x)^2)^3 -
1/((phiM + x + y)^2 + (1/12 phiM^2) (phiM^2 + 4 x*phiM +
4 y*phiM + 12 x*y))^2), {y, 0, Infinity},
Assumptions -> {phiM > 0, x > 0}]


using Mma11, Win7. The result is here but it is huge. Here is an image of a verysmall part of it to make you believe: Nevertheless, one can operate with it. For example,

Plot3D[int, {x, 0, 10}, {phiM, 0, 10}]


gives

Have fun!