Expression involving square roots not simplifying

 expr = (Sqrt[s γ^3] + 2 Sqrt[s γ^7] + Sqrt[s^5 γ^7] + Sqrt[s γ^11] + 
        2 s (Sqrt[s γ^5] + Sqrt[s γ^9]))/(γ (1 + s γ + γ^2)^2);


Simplify[PowerExpand[expr]]

More by way of explanation of the "indifference" that causes Simplify to not budge. In order to factor the expression so that it can be reduced, all the square-roots have to be factored and initially the complexity (computed by Simplify`SimplifyCount, which is equivalent to LeafCount on these examples) remains the same:

Simplify`SimplifyCount[Sqrt[s0^5 γ^7]]
Simplify`SimplifyCount[s0^(5/2) γ^(7/2)]

(*
  11
  11
*)

The actual algorithm used by Simplify is unknown (to me), but it makes sense to reject a transformation that results in an expression with the same complexity as measured by the ComplexityFunction (to avoid getting stuck in an infinite cycle of equivalent-complexity expressions).

While there is a simpler solution (see @Nasser's), another approach is to tweak ComplexityFunction to make the desired steps seem "simpler":

cf = LeafCount[#] + 2 Count[#, Power[_Times, _], {0, ∞}] &; 
Simplify[(Sqrt[s0 γ^3] + 2 Sqrt[s0 γ^7] +
   Sqrt[s0^5 γ^7] + Sqrt[s0 γ^11] + 
    2 s0 (Sqrt[s0 γ^5] + Sqrt[s0 γ^9]))/(γ (1 + s0 γ + γ^2)^2),
 γ > 0 && s0 > 0, ComplexityFunction -> cf]

(*  Sqrt[s0 γ]  *)

Raise the coefficient of Count[] in cf to 5 and the result will be Sqrt[s0] Sqrt[γ].