I want to integrate $$ \int_0^{\infty}dx\,e^{-ax}\frac{1-(2x)^b}{1-2x} $$ where $a,b>0$. My The naive approach was to consider $b$ to be an integer, in which case you get a truncated geometric series above that gives you a sum of gamma functions. Nonetheless, by doing this you get a function that when tried for $b$ non integer gives you a number with an imaginary part, so the naive approach of solving for integer $b$ and hoping that the result can be generalized does not work. Any suggestions?

Any ideas about how to proceed?