Given the function $f(x,y)$ defined as $$f(x,y)\equiv\frac{x^\frac{p}{2}}{\left(1+x+y^2\right)^\frac{p+n}{2}},\ \left\{(p,n)\in\mathbb{Z}^2,x\in\mathbb{R}^+,y\in\mathbb{R} \mid p\ge-1,\ n\ge4\right\},$$ how would you compute $$\underset{f\ge F}{\iint} f(x,y)dydx,\ \left\{F\in\mathbb{R}^+\right\}?$$ In the above expression, $F$ defines the subspace where the function $f(x,y)$ is integrated.
Is there a closed-form solution to this integral? Otherwise, what would be the best way to compute it numerically?
With the attempts I have made so far, I always end up with limits of integration for the inner integral that need to be solved numerically, so that gets me stuck for an analytical closed-form solution. I have not been able to find a change of variable which could avoid the issue? Also the case where $p=-1$ can be tricky at $x=0$, for example, following a trigonometric substitition depending on $x$. Any idea?
Thanks!