Find an antiderivative of the function

Find an antiderivative of the function

Let $u: \mathbb{R}\to \mathbb{R}$ be a function. Find an antiderivative of
$u^2u'''$ in terms of $u$ and its derivatives.
Note: I know an antiderivative of $uu'''$ is $uu''-\frac{u'^2}{2}$.
However it seems for this problem one needs to evaualate antiderivative of
either $uu'u''$ or $u'^3$ which can't be computed easily via integration
by parts