complex analysis - Is there an elementary method for evaluating $displaystyle int_0^infty frac{dx}{x^s (x+1)}$?

I found a way to evaluate $\displaystyle \int_0^\infty \frac{dx}{x^s (x+1)}$ using the assumption that $s\in\mathbb{R}$ and $0

Apparently it should be easily extended to all $s\in\mathbb{C}$ with $0

I posted my solution here: http://thetactician.net/Math/Analysis/Integral1.pdf

I'm pretty sure there's a more concise method for evaluating it...and I'd also like to make the extension to $\mathbb{C}$ more rigorous.

Any ideas?