Ask for different ways to solve $(x+y)dy+(y+1)dx=0$ $(x+y)dy+(y+1)dx=0$
Rewrite the equation to 
$\frac{dx}{dy}+\frac{x}{y+1}=\frac{-y}{y+1}$ 
I can use used the integrating factor $μ(y)=e^{-\int\frac{1}{y+1}dy}$ to solve it.
The answer is $x=\frac{-y^2}{2(y+1)}+\frac{C}{y+1}$
I'm curious that are there any other methods to solve this ODE. I am asking for a different method because sometimes I am not able(or haven't enough time, this one costs me about half an hour to solve it ) to  rewrite the equation into a proper form. So, if I could obtain other methods to solve this kind of ODEs, I may have more chances to solve it in a exam.

Notice, the given D.E. can be easily solved by exact differential form as follows   $$(y+1)dx+(x+y)dy=0$$ Now, comparing the above equation with $Mdx+Ndy=0$, we get $$M=y+1\implies \frac{\partial M}{\partial y}=1$$
$$N=x+y\implies \frac{\partial N}{\partial x}=1$$
since, $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$, hence the given equation is in the exact differential form, hence the solution is given as $$\int_{\text{keeping y constant}} (y+1)dx+\int_{\text{terms free of}\ x}(x+y)dy=C$$
$$ (y+1)\int dx+\int y\ dy=C$$
$$\color{blue}{(y+1)x+\frac{y^2}{2}=C}$$
