So, is your integral
[tex] \int \frac{dr}{\sqrt{\frac{1}{R+r}-\frac{1}{R}}} ? [/tex]
If so, re-write it by first expressing
[tex] \frac{1}{R+r}-\frac{1}{R} [/tex]
as a simple rational expression. You should end up with an integrand of the form
[tex] \sqrt{\frac{ar+b}{cr+d}}. [/tex]