Generalizing the Lambert W function

Question

Can an equation of the form

$ y=e^{\frac{p(y)}{q(y)}}$

be solved for $y$ in terms of the Lambert W function?

$p(x)$ and $q(x)$ are polynomials, for example

$y=e^{\frac{b(y+1)}{y^2+y+\kappa}}.$