Galen-Ashore III

Suppose that $X \sim \text{Beta}(a,b)$ with $\alpha > 1$ and $Y \mid X = x \sim \text{Galenshore}(1-x,\sqrt{x})$. For reference, the PDF of a Galenshore$(\alpha,\theta)$ distribution is

$$f(x) = \dfrac{2}{\Gamma(\alpha)}\theta^{2\alpha}x^{2\alpha - 1}e^{-\theta^2x^2}I_{(0,\infty)}(x)$$

Compute $\mathbb{E}[Y^2]$ when $a = 3$ and $b = 4$.