\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {x}{y \left ( x \right ) +\sqrt {{x}^{2}+1}}}=0} \]
Mathematica: cpu = 0.195525 (sec), leaf count = 104 \[ \text {Solve}\left [\frac {1}{2} \left (\log \left (\frac {-\sqrt {x^2+1} y(x)^2-\left (x^2+1\right ) y(x)+\left (x^2+1\right )^{3/2}}{\left (x^2+1\right )^{3/2}}\right )+\log \left (x^2+1\right )\right )=c_1+\frac {\tanh ^{-1}\left (\frac {3 \sqrt {x^2+1}+y(x)}{\sqrt {5} \left (\sqrt {x^2+1}+y(x)\right )}\right )}{\sqrt {5}},y(x)\right ] \]
Maple: cpu = 0.359 (sec), leaf count = 112 \[ \left \{ -{\frac {4}{3}\ln \left ( 36\,{\frac {\sqrt {{x}^{2}+1}}{y \left ( x \right ) +\sqrt {{x}^{2}+1}}} \right ) }+{\frac {2}{3}\ln \left ( -{\frac {1296}{11} \left ( y \left ( x \right ) \sqrt {{x}^{2}+1} -{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-1 \right ) \left ( y \left ( x \right ) +\sqrt {{x}^{2}+1} \right ) ^{-2}} \right ) }-{\frac { 4\,\sqrt {5}}{15}{\it Artanh} \left ( {\frac {\sqrt {5}}{5} \left ( 3\, \sqrt {{x}^{2}+1}+y \left ( x \right ) \right ) \left ( y \left ( x \right ) +\sqrt {{x}^{2}+1} \right ) ^{-1}} \right ) }+{\frac {2\,\ln \left ( {x}^{2}+1 \right ) }{3}}-{\it \_C1}=0 \right \} \]