\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) = \left ( -x+ \left ( \left ( y \left ( x \right ) \right ) ^{-1}+1 \right ) x+{\it \_F1} \left ( \left ( \left ( y \left ( x \right ) \right ) ^{-1}+1 \right ) x \right ) {x}^{2}-{\it \_F1} \left ( \left ( \left ( y \left ( x \right ) \right ) ^{-1}+1 \right ) x \right ) {x}^{2} \left ( \left ( y \left ( x \right ) \right ) ^{-1}+1 \right ) \right ) ^{-1}=0} \]
Mathematica: cpu = 1.356672 (sec), leaf count = 362 \[ \text {Solve}\left [\int _1^{y(x)} \left (\frac {x \text {$\_$F1}\left (x \left (\frac {1}{K[2]}+1\right )\right )-1}{x K[2] \text {$\_$F1}\left (x \left (\frac {1}{K[2]}+1\right )\right )+x \text {$\_$F1}\left (x \left (\frac {1}{K[2]}+1\right )\right )-K[2]}-\int _1^x \left (\frac {-\frac {K[1] \text {$\_$F1}'\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )}{K[2]}-\frac {K[1] \text {$\_$F1}'\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )}{K[2]^2}+\text {$\_$F1}\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )}{K[1] \left (K[2] \text {$\_$F1}\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )+\text {$\_$F1}\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )\right )-K[2]}-\frac {\left (K[2] \text {$\_$F1}\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )+\text {$\_$F1}\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )\right ) \left (K[1] \left (-\frac {K[1] \text {$\_$F1}'\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )}{K[2]}-\frac {K[1] \text {$\_$F1}'\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )}{K[2]^2}+\text {$\_$F1}\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )\right )-1\right )}{\left (K[1] \left (K[2] \text {$\_$F1}\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )+\text {$\_$F1}\left (K[1] \left (\frac {1}{K[2]}+1\right )\right )\right )-K[2]\right ){}^2}\right ) \, dK[1]\right ) \, dK[2]+\int _1^x \left (\frac {y(x) \text {$\_$F1}\left (\left (\frac {1}{y(x)}+1\right ) K[1]\right )+\text {$\_$F1}\left (\left (\frac {1}{y(x)}+1\right ) K[1]\right )}{K[1] \left (y(x) \text {$\_$F1}\left (\left (\frac {1}{y(x)}+1\right ) K[1]\right )+\text {$\_$F1}\left (\left (\frac {1}{y(x)}+1\right ) K[1]\right )\right )-y(x)}-\frac {1}{K[1]}\right ) \, dK[1]=c_1,y(x)\right ] \]
Maple: cpu = 0.141 (sec), leaf count = 40 \[ \left \{ y \left ( x \right ) ={{\rm e}^{{\it RootOf} \left ( -{\it \_Z}- \int ^{{\frac {x{{\rm e}^{{\it \_Z}}}}{{{\rm e}^{{\it \_Z}}}-1}}}\!{ \frac {1}{ \left ( {\it \_F1} \left ( {\it \_a} \right ) {\it \_a}-1 \right ) {\it \_a}}}{d{\it \_a}}+{\it \_C1} \right ) }}-1 \right \} \]