Definite Integral from 0 to Infinity of x^m over (x^n + a^n)^r

Theorem

 * $\ds \int_0^a \frac {x^m \rd x} {\paren {a^n - x^n}^r} = \frac {\paren {-1}^{r - 1} \pi a^{m + 1 + n r} \, \map \Gamma {\frac {m + 1} n} } {n \sin \frac {\paren {m + 1} \pi} n \paren {r - 1}! \, \map \Gamma {\frac {m + 1} n - r + 1} }$

for $0 < m + 1 < n r$.