$\displaystyle dx/dt-x^3=x$

$\displaystyle dx/dt=(x^3+x)$

$\displaystyle dt=(x^3+x)^{-1}dx$

So I haven't been able to figure out how to do the integral.

I don't think I can do fraction decomposition. At least when I try I don't get anything that makes a whole lot of sense.

I've tried integration by parts and I don't see any thing that looks remotely like the answer. Which is $\displaystyle x= \pm(Ce^{2t}/(1-Ce^{2t}))^{1/2} ,C\geq0$