I'm trying to compute the following integral explicitly.

$$I=\int_{0}^{+\infty} dx \left(1+\frac{1}{x}\right) \frac{\sqrt{x}}{e^{-1}+xe^x}$$

The best I managed to do is to do a change of variable $x=W(y)$, where W is the Lambert function. The integral is then given by

$$I=\int_{0}^{+\infty} \frac{dy}{y} \frac{\sqrt{W(y)}}{e^{-1}+y}$$

Maybe there could be a way to deform the contour of integration in the complex plane and use a residue formula with the new contour as we could have a pole at $y=-e^{-1}$?

I guess we could do the transformation $y=e^x$ and obtain

$$I=\int_{-\infty}^{+\infty} dx \frac{\sqrt{W(e^x)}}{e^{-1}+e^x}$$

The poles would be located at $x=-1\pm i \pi$ but I do not know which contour to choose...

I computed numerically the integral on mathematica which gives $I\simeq 3.9965$