Solving $x+x\ln(x)+\ln(x)=y$ for $x$

Question

For $x,y\in\mathbb{R^+}$ , consider the equation:

$x+x\ln(x)+\ln(x)=y$

with constant $y$,

which is the same as

$x+\ln(x^{x+1})=y$

How do I solve for $x$?

Answer 1

By differentiating, we can discover that $y$ is strictly increasing as a function of $x$, and it's easy enough to see that $y \to -\infty$ as $x \to 0$ and that $y \to \infty$ as $x \to \infty$; so the function is bijective with range $\mathbb{R}$. But I'd be absolutely astonished if there were a closed form for it. Certainly Mathematica couldn't find one.