\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =-{\frac {x}{\sqrt { \left ( y \left ( x \right ) \right ) ^{2}}} \left ( -{x}^{-1}-{\it \_F1} \left ( \left ( y \left ( x \right ) \right ) ^{2}-2\,x \right ) \right ) }=0} \]
Mathematica: cpu = 0.894114 (sec), leaf count = 100 \[ \text {Solve}\left [\int _1^{y(x)} \left (\frac {\sqrt {K[2]^2}}{\text {$\_$F1}\left (K[2]^2-2 x\right )}-\int _1^x \frac {2 K[2] \text {$\_$F1}'\left (K[2]^2-2 K[1]\right )}{\left (\text {$\_$F1}\left (K[2]^2-2 K[1]\right )\right ){}^2} \, dK[1]\right ) \, dK[2]+\int _1^x \left (-\frac {1}{\text {$\_$F1}\left (y(x)^2-2 K[1]\right )}-K[1]\right ) \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.218 (sec), leaf count = 65 \[ \left \{ y \left ( x \right ) =\sqrt {2\,{\it RootOf} \left ( {x}^{2}-2\, \int ^{{\it \_Z}}\! \left ( {\it \_F1} \left ( 2\,{\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+4\,{\it \_C1} \right ) +2\,x},y \left ( x \right ) =-\sqrt {2\,{\it RootOf} \left ( {x}^{2}-2\,\int ^{{\it \_Z}} \! \left ( {\it \_F1} \left ( 2\,{\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+4\,{\it \_C1} \right ) +2\,x} \right \} \]