Integral of an Irrational Function

Calculus Level 5

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

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 \(\text{A}+\text{B}+\text{C}+\text{D}+\text{E}\)

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