Can you use Lambert's W to solve this equation for x?

Question

I would like to solve the following equation for $x$: $$ a= \left( \frac{x}{1-x} \right)^\lambda e^{bx} $$ I'm fairly new to using Lambert's W so I'm not sure if this is an appropriate tool for the problem.

Answer 1

After some simplifications the equation can be transformed into an equation of the form $A + B y = y e^y$ and I'm pretty sure that this equation can't be solved using the Lambert function (unless $A=0$ or $B=0$, of course).

I've solved many equations using the Lambert function and Mathematica always have been able to solve it too. For this one Mathematica wasn't able solve it.