I am using a hierarchical Bayesian model. In one part of it, I have a normal distribution with mean zero and a variance sampled from a Gamma distribution for some hyper-parameters $a_0$ and $a_1$:
$$p(x|\alpha) \sim \mathcal{N}(0,\alpha) $$ $$p(\alpha|a_0,a_1) \sim \text{Gam}(a_0,a_1)$$
Does the distribution of $x$ after integrating $\alpha$ out, i.e., $p(x|a_0,a_1)$, have a closed-form expression?