This is the integral one can find in the Introduction of 'Special Integrals of Gradshteyn and Ryzhik the Proofs - Volume I' by Victor H. Moll:
$$\int_0^{+\infty} \frac{dx}{(1+x^2)^{3/2} \left[ \phi(x) + \sqrt{\phi(x)} \right]^{1/2}}, \quad \phi(x) = 1 + \frac{4x^2}{3(1+x^2)^2} \;.$$ The author doesn't know the final answer. It is claimed that it is $\pi / 2 \sqrt{6}$, though numerical integration contradicts this.
Any ideas how to solve it or where to find clues?
Just out of curiosity, I worked numerically$$I(k)=\int_0^{\infty } \frac{dx}{\left(x^2+1\right)^{3/2} \sqrt{\phi (x)+\sqrt{\phi (x)}}}\qquad, \qquad \phi(x) = 1 + \frac{kx^2}{(1+x^2)^2} $$ and tried to find $k$ such that $$I(k)=\frac{\pi }{2 \sqrt{6}}$$ The closest value I found is $k_*=0.5923509316314110643$ which inverse symbolic calculators did not find any equivalent. $$I(k_*)\approx 0.6412749150809320477747$$ $$\frac{\pi }{2 \sqrt{6}}\approx0.6412749150809320477720$$