Given the integral
$$ \hat{\alpha}({r_{0}})=2\int^{\infty}_{{r_{0}}}\frac{dr}{r \sqrt{1-\frac{2M}{r}} \sqrt{\left(\frac{r}{{r_{0}}}\right)^{2}\left(1-\frac{2M}{{r_{0}}}\right) \left(1-\frac{2M}{r}\right)^{-1}-1}} $$
The solution of the above integral we given here, without shown how, to second order in Eq.(24) as
$$ \hat{\alpha}({r_{0}})=\frac{4M}{r_{0}}+\frac{4M^{2}}{r^{2}_{0}}\left(\frac{15\pi}{16}-2\right)+.... $$
Could someone please show me how to get this result?