I have to compute $\displaystyle\int_{0}^{\infty}\frac{x^{2}}{(x^2+1)(x+1)^{3/2}}\,dx$.
I know that with a change of variables I could make it become the integral of a rational function, but I want expressly to evaluate the integral through contour integration in the complex plane (for example if instead of $3/2$ I had an irrational number, the change of variables wouldn't work).
I was able to compute similar integrals using a keyhole contour that went around the branch points. In those cases, though, the non-integer power was at the numerator, or if at the denominator, it was less than 1. The problem with this one is that I can't bound the function near the branch point in such a way that the small circle integral around it goes to zero as the radius goes to zero. What should I do? Is there a completely different technique to handle this one? Thank you in advance.