Computing $\alpha$ such that $\mathbb{E}[(\tfrac{\alpha}{X + \alpha})^2] = t$, for $t \in (0, 1)$?

Question

Suppose that $X$ is a Binomial random variable with parameters $n, p$. Let $$ F(\alpha) = \mathbb{E}\Big[\big(\frac{\alpha}{X + \alpha}\big)^2\Big], \quad \alpha > 0. $$ Clearly we have the limit relations $$ \lim_{\alpha\to \infty} F(\alpha) = 1, \quad \mbox{and} \quad \lim_{\alpha \to 0} F(\alpha) = 0. $$ Let $t \in (0, 1)$. Is there a way to compute $F^{-1}(t)$? Equivalently, $\alpha^\star(t)$ that solves $$ F(\alpha) = t. $$ The issue is that I do not see any way to really simplify the expression for $F(\alpha)$.