As part of a project related to kinematic fluid dynamics, the following integral appeared in the moment expansion $$ \int_0^{\alpha} t^m\,_2F_0\left(\begin{matrix}-\ell-\frac{1}{2},m+2\\-\end{matrix};\frac{t-1}{\epsilon}\right)dt $$ Now while I have no illusions that this has a closed form, it is sufficient to be able to evaluate it numerically. Since $\alpha$ is usually positive and not large compared to 1, a numeric integration should be fairly efficient provided there is an efficient way to calculate $_2F_0$. Unfortunately, I can't seem to find good routines to do that. Most special function packages have the related function $_1U_1$, but that often requires evaluating it at very large arguments, raising concerns about accuracy.

Does anyone know of a good numerical method for calculation $_2F_0$ directly?