$\displaystyle \displaystyle \int{x^3e^{x^2}\,dx} = \int{\frac{1}{2}x^2\cdot 2x\,e^{x^2}\,dx}$.
Now use integration by parts with $\displaystyle \displaystyle u = \frac{1}{2}x^2 \implies du = x\,dx$ and $\displaystyle \displaystyle dv = 2x\,e^{x^2}\,dx \implies v = e^{x^2}$ and the integral becomes
$\displaystyle \displaystyle \frac{1}{2}x^2e^{x^2} - \int{x\,e^{x^2}\,dx}$
$\displaystyle \displaystyle = \frac{1}{2}x^2e^{x^2} - \frac{1}{2}\int{2x\,e^{x^2}\,dx}$
$\displaystyle \displaystyle = \frac{1}{2}x^2e^{x^2} - \frac{1}{2}e^{x^2} + C$.