Fisher Information for Beta Distribution

Question

I am trying to find the Fisher Information for $\operatorname{Beta}(\alpha,2)$. I used the following approach: $$f_X(\alpha,2)=\alpha(\alpha+1)x^{(\alpha-1)}(1-x),\alpha>0$$ $$ L(\alpha|x_i)=\alpha^n(1+\alpha)^n\prod_{i=1}^nx_i^{(\alpha-1)}\prod_{i=1}^n(1-x_i)$$ $$\ell(\alpha|x_i)=n(\ln{(\alpha)}+\ln(1+\alpha))+(\alpha-1)\sum_{i=1}^n\ln(x_i)+\sum_{i=1}^n\ln(1-x_i)$$ $$\frac{\partial}{\partial\alpha}\ell(\alpha|x_i)=\frac{n}{\alpha}+\frac{n}{\alpha+1}+\sum_{i=1}^n\ln(x_i)$$ $$\frac{\partial^2}{\partial\alpha^2}\ell(\alpha|x_i)=-\frac{n}{\alpha^2}-\frac{n}{(1+\alpha)^2}$$ $$I_n(\alpha)=-E\left[\frac{\partial^2}{\partial\alpha^2}\ell(\alpha|x_i)\right]=\frac{n}{2\alpha^2+2\alpha+1}$$ This would seem to imply that a lower bound on the variance for estimators of $\alpha$ is given by $I_n(\alpha)^{-1}=\frac{2\alpha^2+2\alpha+1}{n}$. This value, however, is giving me trouble in follow-on calculations. I am asking if my work so far is correct, and if not, what I missed. Thank you.