Let $v,q,b ∈ \mathbb R^+ $ and $m∈ \mathbb R^+ ∪$ {0} , then how do we evaluate the definite integral $$\int_m^\infty \frac {x^{v+r-1}}{(x^v-m^v+q)^{b+1}}dx$$ , where $r ∈ \mathbb N $ ? At least a solution(exposition) for $r=1$ will be appreciated.
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The integral converges when $r\lt bv$. When $m=0$, the change of variable $x^v=q(1-u)/u$ yields that the integral to be computed is $$ q^{1+r/v}\int_0^1u^{b-r/v-1}(1-u)^{r/v}\mathrm du=q^{1+r/v}\mathrm{B}(b-r/v,r/v+1). $$ When $m\gt0$, the change of variable $x^v=m^v(1+u)$ yields that the integral to be computed is $$ v^{-1}m^{r-bv}a^{b+1}F(a,b,r/v),\qquad a=q^{-1}m^v, $$ where, for every positive $(a,b,c)$ such that $b\gt c$, $$ F(a,b,c)=\int_0^\infty(1+u)^c(1+au)^{-b-1}\mathrm du. $$ Various values of $F(a,b,c)$ are direct, for example $F(0^+,b,c)=+\infty$, $F(a,+\infty,c)=0$, $F(1,b,c)=1/(b-c)$, $F(a,b,0^+)=1/a$. The function $F$ itself could be a simple transform of some twisted beta-like integral.
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