Volume of $x \in \mathbb{R}^n$ satisfying both $\prod_{i=1}^n {x_i}^2 < \alpha$ and $\sum_{i=1}^n {x_i}^2 < \beta$

Question

I want to find the volume of the $x \in \mathbb{R}^n$ satisfying both ${x_1}^2 \cdot {x_2}^2 \cdot \ldots \cdot {x_n}^2 < \alpha$ and ${x_1}^2 + {x_2}^2 +\ldots + {x_n}^2 < \beta$. I can solve this easily for $n=2$, and possibly $n=3$ with some effort, but I'm not sure how to get an analytic solution in general? If an analytic solution is too hard, then it'd be helpful to know whatever can be said about the scaling of this volume with $n$, $\alpha$, and $\beta$. Thank you!