I have problem of solving or approximating the following integral:

$$ \int_{0}^{\infty} \frac{(1- (ax+1)^{(1-n)})^{(m-1)}}{(ax+1)^n} dx$$

I tried substitution or simplification, but it did not work. It was not successful.

Can anyone suggest any tips please?

Thank you.


I am assuming $a>0,n>1$.

Keep $1-(ax+1)^{1-n}=u,du=a(n-1)dx/(ax+1)^n$ which gives$$I=\int_{0}^1\frac{u^{m-1}}{a(n-1)} du=\begin{cases}\frac1{am(n-1)},&m>0\\\infty,&m\le0\end{cases}$$

For $n=1$ the integrand is $0$. For $n<1$ we get $-\infty$ for $m\le0$ but the same result as above for $m>0$.


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