\[ y'(x)=2 x-F(x) \left (-x^4+2 x^2 y(x)-y(x)^2+1\right ) \] ✓ Mathematica : cpu = 0.461554 (sec), leaf count = 48
\[\left \{\left \{y(x)\to -\frac {2 e^{\int _1^x 2 F(K[5]) \, dK[5]}}{e^{\text {Integrate}[2 F(K[5]),\{K[5],1,x\},\text {Assumptions}\to \text {True}]}-2 c_1}+x^2+1\right \}\right \}\]
✓ Maple : cpu = 0.76 (sec), leaf count = 44
\[ \left \{ y \left ( x \right ) ={\frac {-{x}^{2} \left ( {{\rm e}^{\int \!F \left ( x \right ) \,{\rm d}x}} \right ) ^{2}+{\it \_C1}\,{x}^{2}+ \left ( {{\rm e}^{\int \!F \left ( x \right ) \,{\rm d}x}} \right ) ^{2}+{\it \_C1}}{- \left ( {{\rm e}^{\int \!F \left ( x \right ) \,{\rm d}x}} \right ) ^{2}+{\it \_C1}}} \right \} \]