 Solution

Integrate the equation without dividing by $y$:This holds on a connected interval excluding $x=-1$. The real solutions are differentiable branches of this implicit curve. Where the radicand is positive, $y=\pm \sqrt{2x-2\log |x+1|+C}$, with a locally fixed sign.

A zero of $y$ can belong to a classical solution only at $x=0$, since the original equation then requires $x/(x+1)=0$. For $C=0$ near zero, $2x-2\log (1+x)=x^2+O(x^3)$; the smooth continuations have slopes $+1$ or $-1$ and switch between square-root signs across zero. Keeping the principal nonnegative square root across zero instead gives a cusp. Thus the implicit expression requires a smooth branch choice, not an unrestricted global choice of one square-root sign.