Consider the integral $$\int_0^1\int_{0}^y (y-x)(xy)^{a-\frac{3}{2}} [(1-x)(1-y)]^{b-\frac{3}{2}}\,dx\,dy$$
The answer is $$\frac{4\Gamma(2a-1)\Gamma(2b-1)}{\Gamma(2a+2b-1)}$$ where $\Gamma(a)$ is Gamma function. The result can be verified numerically by choosing some $a$ and $b$. I wonder how to prove this fact.
This result is related to an extension of Beta distribution on 2d plane.