Central chi-squared distribution with $\nu$ degrees of freedom, that is$\chi^2_\nu$ ($\nu \in\mathbb N$), only has an elementary closed form CDF for each even value of $\nu$, and that doesn't even mean that a generic elementary expression can be found involving all cases, but that an elementary expression exists for each even $\nu$. In any case, there's not an elementary expression for odd values of $\nu$. This is the case for central chi-squared, let alone the more generic non-central chi-squared.
Anyway, closed form is always an ambiguous concept. For instance, using incomplete gamma function there is a closed expression in the central case (and maybe in the non-central case too) but this is almost as general as using the very same definition of CDF (with an integral from $-\infty$ to a certain point) as such a closed form expression.