How to find $int_{0}^{infty} log(1+x)(x^a)exp(-bx) d_x$ step by step?

How can I find $int_{0}^{infty} log_{2}(1+x)x^aexp(-bx) d_x$ step by step?

I’ve got a Mathematica solution, but I’d like to know and understand the steps taken to reach the solution.

Thanks!