The value of the integeral \(\int_0^1 \frac{\sqrt{x} d x}{(1+x)(1+3 x)(3+x)}\) is: [JEE Main 2021, 27 Aug (Shift 2)]

\(\frac{\pi}{4}\left(1-\frac{\sqrt{3}}{6}\right)\)

\(\frac{\pi}{8}\left(1-\frac{\sqrt{3}}{2}\right)\)

\(\frac{\pi}{4}\left(1-\frac{\sqrt{3}}{2}\right)\)

\(\frac{\pi}{8}\left(1-\frac{\sqrt{3}}{6}\right)\)