\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =-F \left ( x \right ) \left ( - \left ( y \left ( x \right ) \right ) ^{2}+2\,{x}^{2}y \left ( x \right ) +1-{x}^{4} \right ) +2\,x=0} \]
Mathematica: cpu = 0.474060 (sec), leaf count = 49 \[ \left \{\left \{y(x)\to \frac {e^{\int _1^x 2 F(K[5]) \, dK[5]}}{c_1-\frac {1}{2} e^{\text {Integrate}[2 F(K[5]),\{K[5],1,x\},\text {Assumptions}\to \text {True}]}}+x^2+1\right \}\right \} \]
Maple: cpu = 0.359 (sec), leaf count = 46 \[ \left \{ y \left ( x \right ) ={1 \left ( {\frac {{\it \_C1}\,{x}^{2}}{ \left ( {{\rm e}^{\int \!F \left ( x \right ) \,{\rm d}x}} \right ) ^{2}} }-{x}^{2}+{\frac {{\it \_C1}}{ \left ( {{\rm e}^{\int \!F \left ( x \right ) \,{\rm d}x}} \right ) ^{2}}}+1 \right ) \left ( {\frac {{\it \_C1}}{ \left ( {{\rm e}^{\int \!F \left ( x \right ) \,{\rm d}x}} \right ) ^{2}}}-1 \right ) ^{-1}} \right \} \]