Are $x$ and $y$ supposed to be integers ? If not, I am afraid that you could fac gaussian hypergeometric functions.
– Claude LeiboviciDec 10 '21 at 04:53
2
https://math.stackexchange.com/questions/3419834/integrals-related-to-the-reciprocal-beta-function?noredirect=1
– SvyatoslavDec 10 '21 at 05:20