When I try and simplify this expression:
In[1]:= FullSimplify[Sqrt[ x^2 + 2 y^2 + 2 y Sqrt[x^2 + y^2] ], (x | y) \[Element] Reals]
Out[1]= Sqrt[x^2 + 2 y (y + Sqrt[x^2 + y^2])]
it does not simplify properly.
However, Mathematica knows it equals a simpler expression:
In[2]:= FullSimplify[ Sqrt[x^2 + 2 y^2 + 2 y Sqrt[x^2 + y^2] ] == Sqrt[x^2 + y^2] + y, (x | y) \[Element] Reals]
Out[2]= True
Why doesn't Mathematica simplify this expression, and how can I make it do it anyway, even if the expression is part of a larger expression?