As a direct answer to your last question, substituting $x=(1-t)/t$ in your integral gives $$\int_0^{1/(1+a)}(1-t)^st^{n-s-2}\,dt$$ which is by definition an incomplete beta integral. That is a perfectly good answer, you can't expect an expression in terms of elementary functions if all your parameters are general.

As a direct answer to your last question, substituting $x=(1-t)/t$ in your integral gives $$\int_0^{1/(1+a)}(1-t)^st^{n-s-2}\,dt$$ which is the standard definition of an incomplete beta integral. That is a perfectly good answer, you can't expect an expression in terms of elementary functions if all your parameters are general.