\[ y'(x)=2 x-F(x) \left (-x^4+2 x^2 y(x)-y(x)^2+1\right ) \] ✓ Mathematica : cpu = 0.468694 (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 = 0.589 (sec), leaf count = 46
\[ \left \{ y \left ( x \right ) ={1 \left ( {\frac {{\it \_C1}\,{x}^{2}}{ \left ( {{\rm e}^{\int \!F \left ( x \right ) \,{\rm d}x}} \right ) ^{2}}}-{x}^{2}+{\frac {{\it \_C1}}{ \left ( {{\rm e}^{\int \!F \left ( x \right ) \,{\rm d}x}} \right ) ^{2}}}+1 \right ) \left ( {\frac {{\it \_C1}}{ \left ( {{\rm e}^{\int \!F \left ( x \right ) \,{\rm d}x}} \right ) ^{2}}}-1 \right ) ^{-1}} \right \} \]