It doesn't look like it can be solved exactly...
solve[ 1/( Sqrt[2+x] - Sqrt[x]) + 1/( Sqrt[2-x] + Sqrt[x] ) = Sqrt[2]/x] - Wolfram|Alpha
It doesn't look like it can be solved exactly...
solve[ 1/( Sqrt[2+x] - Sqrt[x]) + 1/( Sqrt[2-x] + Sqrt[x] ) = Sqrt[2]/x] - Wolfram|Alpha