Integrating $x^3 \sqrt{x^2+1} $ I need to solve this integral $\int x^3 \sqrt{x^2+1} dx$
Someone could explain to me how?
I have tried to use substitutions, i.e. $x^2+1$ and  $\sqrt{x^2+1}$; but seems like its not correct. Someone could explain to me how?

By substituting $x=\sinh t$ we have:
$$ I = \int \sinh^3 t\cosh^2 t\,dt =\int \sinh^3 t\,dt + \int\sinh^5 t\,dt.$$
Just turn $\sinh t$ into $\frac{e^t-e^{-t}}{2}$ and integrate termwise what you get through the binomial theorem.
