How to integrate the multiplication of a polynonmial, a fraction and an exponential

Question

EDITED:

Any ideas on how to do the integral of this function?

$\int_0^1\dfrac{x^2}{(x+2)(x-2)}e^\frac{x}{x+2}e^\frac{x}{x-2}~dx$

Answer 1

This is not a complete answer:

Substitution of

$z=\frac{2 x^2}{(x-2) (x+2)}$

and the settings

$\kappa =-\frac{1}{3 \sqrt{3}}$, $u=-\frac{2}{3}$, $\nu =\frac{3}{2}$ and $\mu =-\frac{1}{2}$

leads to the more general form of the integral above:

$I = \kappa \int_0^1 e^{u\, z} z^{\nu -1} \left(1-\frac{u}{2} \,z\right)^{\mu -1} \, dz$.

Now you can figure out the solution. In fact you have a special function (Hypergeometric, Meijer G or H-Fox-function with two variables). Reduction takes place for $u/2->u$ or in the case:

$I =\kappa \int_0^\infty e^{u\, z} z^{\nu -1} \left(1-\frac{u}{2} \,z\right)^{\mu -1} \, dz=2^{\nu } (-u)^{-\nu } \text{Gamma}[\nu ]\, \text{HypergeometricU}[\nu,\mu +\nu ,2]$

EDIT

$e^{\frac{x}{(x-2)}}\,e^{\frac{x}{(x+2)}}=e^{\frac{2\,x^2}{(x-2) (x+2)}}$