 Mellin integral of a reciprocal binomial

For real $p>0$ and $0<s<p$, the displayed improper integral converges: its endpoint powers are $x^{s-1}$ near zero and $x^{s-p-1}$ near infinity. The change of variables $t=x^p$ givesHere the beta function, the beta-gamma identity, and the Gamma reflection formula evaluate the integral. The same formula holds for complex $s$ in the Mellin fundamental strip $0<\mathrm{Re}s<p$, by absolute convergence and analytic continuation within the strip. For integer $p$, a sector contour of angle $2\pi /p$ gives a direct residue theorem proof, with a consistent branch of $z^{s-1}$.