Evaluation of an Integral

The Moll-Arias de Reyna integral [1] $${\int }_0^{\mathrm{\infty }}\frac{dx}{(x^2+1{)}^{3\mathrm{/}2}}\frac{1}{\sqrt{\phi (x)+\sqrt{\phi (x)}}}$$ $$\phi (x)=1+\frac{4}{3}{\left(\frac{x}{x^2+1}\right)}^2$$ is generalised and several values are given.