Can the polynomials $p(x)$ with $deg(p)=3$, for which $\sqrt{p(x)}$ has an elementary antiderivate, be somehow classified ?
According to Wolfram alpha, already for $$p(x)=(x-1)(x-2)(x-3)$$ , no elementary antiderivate exists, whereas the cases $$p(x)=(x-a)^3$$ and $$p(x)=(x-a)^2(x-b)$$ lead to an elementary antiderivate. However, those cases are very special because of $\sqrt{(x-a)^2}=|x-a|$.
Are those special cases the only cases, or can an elementary antiderivate exist for other $p(x)$ ?