I consider the following integral: $$ I(z,s)=\int_0^\infty \frac{x^{z-1}}{(P(x))^s}dx, $$ where $P(x) = a_0 + a_1 x + \cdots + a_n x^n$ is a polynomial of degree $n \geq 2$ with $P(x) > 0$ for $x \geq 0$. Here $z$, $s$ are complex-valued variables. My question is what are the particular cases of $P$ when the integral $I(z,s)$ can be written via known special functions?
The simpliest case when it is true is $P(x) = (c_0 + c_1 x)^n$, $c_0$, $c_1 > 0$, so that $$ I(z,s) = c_0^{-ns+z}c_1^{-z}B(z,ns-z). $$
Another case is when $P(x) = a_0 + a_n x^n$, $a_0$, $a_n>0$, so that $$ I(z,s) = \frac 1 n a_0^{-s+z/n} a_n^{-z/n} B\biggl(\frac z n,s-\frac z n\biggr). $$
Or, more generally, $P(x) = (c_1+c_2 x^k)^l$, $c_1$, $c_2 > 0$, $kl = n$. Are there some other, less trivial cases?