I must explicitly solve the following ODE

$\displaystyle rf''(r)+f'(r)+f'(r)^3=0$.

I used partial fractions to obtain the equation

$\displaystyle \ln f'(r)+\frac{1}{f'(r)}-\ln(f'(r)+1)=\ln r+k$

for some constant $\displaystyle k$. However, I'm not sure this helps. Any hints or direction would be much appreciated. Thanks!