"Closed form" solution of $x(e^x+a) = b$

Question

Is there a way to express the solution of $$x(e^x+a) = b$$ for $a,b, x \ge 0$ and in terms of more or less standard functions?

I already know that if $a=0$, then $x = W(b)$, where $W$ is the Lambert W function. I tried to get this in a form which can be solved by the W function, but unfortunately it does not seem to work. Maybe there is another approach, or possibly its just not possible.

Maybe you could help?

Answer 1

From a formal point of view, there is a solution in terms of the generalized Lambert function writing

$$e^{-x}=\frac{x}{b-a x}$$ (have a look at equation $(4)$ in the linked paper).

This being said, it is not very practical and think about numerical methods.