My friend posed to me an interesting physics problem that reduces to the differential equation $\frac{dy}{dx} = \frac{a}{y} + b$ where $a$ and $b$ are known, nonzero constants. After reading up on how to solve differential equations like this, I got it down to $\frac{by - a\space \ln (a + by)}{b^2} = x + C$ and this is where I hit a dead-end. Wolfram|Alpha tells me that $y = \dfrac{a\left(-W\left(-\dfrac{e^{\dfrac{(x+C)b^2}{a}-1}}{a}\right)\right)-a}{b}$ where $W(z)$ is the Lambert W-function, but this is incredibly complex and I have no idea how it got to that point.
How one would get to this solution, and is there possibly a simpler way of expressing this?