Why doesn't Mathematica simplify a square root of an expression that equals a square of a positive real?

Change to polar coordinates, {x -> r Cos[t], y -> r Sin[t]}:

FullSimplify[ Sqrt[2 y Sqrt[x^2 + y^2] + x^2 + 2y^2] /. 
 {x -> r Cos[t], y -> r Sin[t]}, r > 0 && 0 < t< 2Pi]

r (Sin[t] + 1)

This should simplify your long expression ...

Possibly because the second simplification request is much easier than the first. Consider:

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Also, the second FullSimplify can be done by Simplify. Not surprising since all MMA has to do is square both sides.

edit: Perhaps this will suggest an approach:

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You might do this:

Simplify[Sqrt[x^2 + 2 y^2 + 2 y Sqrt[x^2 + y^2]] /. x^2 -> u^2 - y^2, {u > 0, y > 0}] /. u -> Sqrt[x^2 + y^2]

The result is here: y + Sqrt[x^2 + y^2]

I required that both u and y are positive. If they are not:

Simplify[Sqrt[x^2 + 2 y^2 + 2 y Sqrt[x^2 + y^2]] /.x^2 -> u^2 - y^2, {u < 0, y < 0}] /. u -> Sqrt[x^2 + y^2]

You get another result: Abs[-y + Sqrt[x^2 + y^2]]

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