1\int \frac{dx}{ 1 + \sqrt{x} \sqrt{x}.\sqrt{1 + x}}
x = tan^ 2 \theta
dx = 2tan \theta sec ^ 2\theta d\theta
integral becomes
\int \frac{ 2 tan \theta sec ^ 2 \theta }{ 1 + \sqrt{tan ^ 2\theta } . \sqrt{tan ^ 2\theta .\sqrt{1 + tan ^ 2\theta }}}
2\int \frac{sec \theta d\theta }{ 1 + tan \theta }
2\int \frac{\frac{1}{cos \theta } d\theta }{ 1 + \frac{sin \theta }{cos \theta }}
2\int \frac{ d\theta }{sin \theta + cos \theta }
