I am dealing with an expression containing up to third derivatives with respect to the parameters of the hypergeometric function of the form
$_2F_1(\{-\frac{n_1}{2},\frac{1}{2}-\frac{n_1}{2}\},\{1-2 n_2\},x)$
where $n_1$ and $n_2$ are non-negative integers. The first of these for $n_1\le 3$ and $n_2=0$ can all be evaluated using the HypExp2 package and Mathematica 12 in terms of logarithms and PolyGamma functions. But as soon as $n_1\ge 4$ or $n_2\ne 0$ examples appear that fail to be evaluated. For example the derivative
$_2F^{((2,0),(1),0)}_1(\{-2,-\frac{3}{2}\},\{-1\},x)$
takes a long time to evaluate and in the end produces internal variables of the HypExp2 package which do not cancel out. Mathematica 12 without the package does not even give numerical values unless x=0. Is there a way to evaluate derivatives of this type in closed form?