Your existing code appears to produce a solution (albeit large) on Mathematica 8
sol = DSolve[{p'[r] ==
1/((r^2)*(((R - S)/(R^3)) - (1/(r^2)*(1 - S/r)))^(1/2))}, p[r], r]
You can add the additional constraints on the solution, as part of the simplification. It doesn't appear to make a significant difference. Were you expecting something different?
Simplify[sol, {S, R} \[Element] Reals && S > 0 && R > 3*sqrt (3)*S/2]
Minor Correction
FullSimplify[sol, {S, R} \[Element] Reals && S > 0 && R > 3*sqrt (3)*S/2]
Appears to simplify some of the terms, but only a little.