$n$ is a positive integer, and $\mu$ is a real number. Observations $X_1,...X_n$ are independent and each of them follows $$f(x;\mu)=\begin{eqnarray} \left\{ \begin{array}{l} \frac{\mu}{\sqrt{2\pi x^3}}\exp\{-\frac{1}{2}(\sqrt{x}-\frac{\mu}{\sqrt{x}})^2\} \quad(x\gt0) \\ 0 \quad(x\leq0) \end{array} \right. \end{eqnarray}$$
$\alpha=\frac{a}{\mu}+\frac{a}{\mu^2}$
What is a MLE of $\alpha$, $\hat{\alpha}$?
I am allowed to use $S_n=\left(\frac{1}{n} \sum_\limits{i=1}^n \frac{1}{X_i}\right)^{-1}$.
What I have tried
$$L=\frac{\mu^n}{(2\pi x^3)^\frac{n}{2}}\exp\{-\frac{1}{2}\displaystyle \sum_{i=1}^n(\sqrt{x_i}-\frac{\mu}{\sqrt{x_i}})^2\} \\\ l=n\log \mu -\log(2\pi x^3)^\frac{n}{2}+\{-\frac{1}{2}\displaystyle \sum_{i=1}^n(\sqrt{x_i}-\frac{\mu}{\sqrt{x_i}})^2\} $$
and led $$\mu=\frac{1+\sqrt{1+4\alpha}}{2\alpha}$$
Then I substituted it $$l=n\log \frac{1+\sqrt{1+4\alpha}}{2\alpha} -\log(2\pi x^3)^\frac{n}{2}+\{-\frac{1}{2}\displaystyle \sum_{i=1}^n(\sqrt{x_i}-\frac{\frac{1+\sqrt{1+4\alpha}}{2\alpha}}{\sqrt{x_i}})^2\} \\\ = n\log (1+\sqrt{1+4\alpha})- n \log{2\alpha} -\log(2\pi x^3)^\frac{n}{2}+\{-\frac{1}{2}\displaystyle \sum_{i=1}^n(\sqrt{x_i}-\frac{\frac{1+\sqrt{1+4\alpha}}{2\alpha}}{\sqrt{x_i}})^2\}$$
but I could not get to the conclusion after setting this. $$\frac{\partial l}{\partial \alpha}=0 \\\ $$