integral with log and exponential

Question

This integral seems to complex for me and I could not find a solution. Can Laplace transform be a useful? Is anyone willing to help me? Thanks.

$\int_{0}^{\infty} (\frac{log(1+x)e^{(-x/b)}}{ax+c})dx$

Answer 1

I do not know if I should be able to finish but this could be a way using Feynman trick.

Let us consider $$I(k)=\int_0^\infty \frac{ \log (1+kx)}{a x+c}e^{-\frac{x}{b}}\,dx$$ $$I'(k)=\int_0^\infty \frac{x e^{-\frac{x}{b}}}{(k x+1) (a x+c)}\,dx$$ $$\frac{x }{(k x+1) (a x+c)}=\frac 1{a-ck}\left(\frac{1}{k x+1}-\frac{c}{a x+c} \right)$$ $$(a- ck)I'(k)=\frac{c e^{\frac{c}{a b}} }{a}\text{Ei}\left(-\frac{c}{a b}\right)-\frac{e^{\frac{1}{b k}} }{k}\text{Ei}\left(-\frac{1}{b k}\right)$$ This could be expressed in terms of the incomplete gamma function.

Going to $I(k)$ leaves us with a simple integral and a nasty one but it could be easy to perform numerical integration of it (this would be much easier to do than for the initial integral).