Integral of an Irrational Function

$\text{I} = \int_{0}^{1} \dfrac{\sqrt{x}\sqrt{1-x}}{(1+x)(x^2+1)} \mathrm{d}x$

If $\text{I}$ can be expressed as $\displaystyle \dfrac {{\pi}^{A}}{\sqrt {2}} \left(\sqrt{\sqrt{\text{B}}} \cos\left(\dfrac{\text{C}\pi}{\text{D}}\right) - \text{E} \right)$ where $\text{A},\text{B},\text{C},\text{D}$ and $\text{E}$ are positive integers, $\gcd (\text{C},\text{D}) =1$ and $\text{B}$ is a prime number.

Evaluate $\text{A}+\text{B}+\text{C}+\text{D}+\text{E}$