TITLE: Solving the integral $\int_0^1\int_0^1\frac{1}{(1-\alpha x^\beta y^\gamma)^2}dxdy$
QUESTION [11 upvotes]: How to solve
$$\int_0^1\int_0^1\frac{1}{(1-{\alpha}x^{\beta}y^{\gamma})^2}dxdy$$
where $0\leq\alpha<1$, $\beta\geq0$, and $\gamma\geq0$? Does it have any closed-form? 
I'm not getting any method to solve this integral. Any hint is appreciated.

REPLY [4 votes]: One feels slightly sorry to post this in view of the truly Herculean task that a previously posted answer represents, but it happens that, at least for every $\alpha$ in $[0,1)$ and every nonnegative $\beta$ and $\gamma$, 

$$\int_0^1\!\!\int_0^1\frac{dxdy}{(1-{\alpha}x^{\beta}y^{\gamma})^2}=\sum_{n=0}^\infty\frac{n+1}{(n\beta+1)(n\gamma+1)}\alpha^n$$

Two remarks:

This identity actually holds for every complex parameters such that $|\alpha|<1$, $\Re\beta>0$ and $\Re\gamma>0$.
Various equivalent presentations of the RHS exist, as series and/or as special functions evaluated at $\alpha$, but these alternative formulas/denominations seem to bring no further mathematical understanding of the situation hence we omit them.

The proof of the identity above is a one-liner: integrate term by term the series expansion $$\frac1{(1-{\alpha}x^{\beta}y^{\gamma})^2}=\sum_{n=0}^\infty(n+1)\alpha^nx^{n\beta}y^{n\gamma}$$