\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =-{\frac {-xy \left ( x \right ) -y \left ( x \right ) +\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}{x}^{2}-x\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}y \left ( x \right ) }{x \left ( 1+x \right ) }}=0} \]
Mathematica: cpu = 0.152519 (sec), leaf count = 135 \[ \left \{\left \{y(x)\to \frac {x \left (-2 (x+1)^{\sqrt {2}} e^{\sqrt {2} c_1+\sqrt {2} x}+e^{2 \sqrt {2} c_1+2 \sqrt {2} x}-(x+1)^{2 \sqrt {2}}\right )}{2 (x+1)^{\sqrt {2}} e^{\sqrt {2} c_1+\sqrt {2} x}+e^{2 \sqrt {2} c_1+2 \sqrt {2} x}-(x+1)^{2 \sqrt {2}}}\right \}\right \} \]
Maple: cpu = 0.172 (sec), leaf count = 55 \[ \left \{ \ln \left ( 2\,{\frac {x \left ( \sqrt {2\, \left ( y \left ( x \right ) \right ) ^{2}+2\,{x}^{2}}+y \left ( x \right ) +x \right ) }{y \left ( x \right ) -x}} \right ) +x\sqrt {2}-\ln \left ( x \right ) - \sqrt {2}\ln \left ( 1+x \right ) -{\it \_C1}=0 \right \} \]