How to solve equation $(ax+b)\exp(x) = (cx+d)$ using Lambert W function

Question

Are there any closed form solutions to

$$(ax+b)\exp(x)= cx+d$$

for real-valued $a,b,c$ and $d$?

Answer 1

Rewrite the equation as $$e^{-x}=\frac{a x+b}{c x+d}=\frac a c\,\frac{x+\frac b a}{x+\frac d c}$$

The solution is given in terms of the generalized Lambert function (have a look at equation $(4)$ in the paper).