I have a Beta random variable $X \sim \text{Beta}(\alpha, \beta)$, and I'm interested in $\mathbb{E}[e^{2X}]$.
The Beta distribution moment generating function is
$$f(t) = {\displaystyle 1+\sum_{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}} = {}_1F_1(\alpha, \alpha + \beta; t)$$
So I need to compute $f(2)$. But I need to do that quickly for many pairs $(\alpha, \beta)$. What is an efficient method to compute this?