calculation of $\int^{1}_{0}\frac{1}{\sqrt{x-x^2}(x^2+3x+2)}dx$ 
Calculation of $$\int^{1}_{0}\frac{1}{\sqrt{x-x^2}(x^2+3x+2)}dx$$

My Try: Let $$I = \int^{1}_{0}\frac{1}{\sqrt{x-x^2}(x+1)(x+2)}dx$$
Put $x=\sin^2 \theta$ and $dx = 2\sin \theta \cos \theta$ and changing limits
So $$I = \int^{\frac{\pi}{2}}_{0}\frac{2}{(1+\sin^2 \theta)(2+\sin^2 \theta)}d\theta$$
Could someone help me to solve it , Thanks

Hint. Note that
$$\frac{2}{(1+\sin^2 \theta)(2+\sin^2 \theta)}
=\frac{2}{1+\sin^2 \theta}-\frac{2}{2+\sin^2 \theta}.$$
Moreover, for $k>0$,
$$\frac{1}{k+\sin^2 \theta}=\frac{\frac{1}{k}d(\tan  \theta)}{1+\frac{(k+1)\tan^2 \theta}{k}}.$$
Finally let $t=\frac{\sqrt{k+1}\tan \theta}{\sqrt{k}}.$
