A integral about Powers of x and binomials.

Question

Here is the integral $$\int_0^\infty {\frac{{{x^{p - 1}}}}{{x + a}}{{\left( {bx + c} \right)}^q}} dx,where{\text{ }}a,b,c > 0,p,q \geqslant \frac{1}{2}$$

Answer 1

As already stated above, since the harmonic series diverges, one necessary convergence condition is $\Big[(p-1)+q\Big]-1<-1\iff p+q<1$, for $x\to\infty$. For $x\to0$, we have $(p-1)>-1\iff$ $p>0$. So for $q\geqslant\dfrac12$ the integral diverges. But if we were to abandon this latter demand, we'd have

$$I=\frac\pi a\cdot\csc\Big[(p+q)\pi\Big]\cdot\bigg[a^p\cdot(ab-c)^q-\frac{c^{p+q}}{b^p}\cdot\frac{\Gamma(p)}{\Gamma(-q)}\cdot\frac{_{_2}F_{_1}\bigg(1,~p~;~p+q+1~;~\dfrac c{ab}\bigg)}{(p+q)!}\bigg]$$

where $_{_2}F_{_1}$ is the Gaussian hypergeometric function. This formula is deduced in a similar manner to that from your previous question.