I have the following integral $$ \int_0^1 \int_0^1 \frac{(y-1)^3}{(a(x-1) x y +b (y-1))^2} dx dy $$
and I know that this integral can also be expressed as
$$ 2\int_0^1 \int_0^1 \frac{x^3 y^3 (x-1)}{(a(x-1) (y-1) -b x (1-xy))^2} dx dy $$
I need to find a change of variables that transforms one into the other.
Comments: To keep the same limits of integration, this type of change of variables work $$ x\to (1-x')^n ~,~~y\to (1-y')^n $$
However, I think that the needed change of variables involves both variables, such as $$x\to f(x',y')$$ $$y\to g(x',y')$$, and then, with what I have tried, I do not recover the same limits. Any suggestion, or comment about where to start would be very much appreciated.