\[ y'(x)=\frac {y(x)}{x \log (x)}-F(x) \left (-y(x)^2-2 y(x) \log (x)-\log ^2(x)\right ) \] ✓ Mathematica : cpu = 1.40908 (sec), leaf count = 73
DSolve[Derivative[1][y][x] == y[x]/(x*Log[x]) - F[x]*(-Log[x]^2 - 2*Log[x]*y[x] - y[x]^2),y[x],x]
\[\left \{\left \{y(x)\to \frac {\int _1^x\frac {F(K[5])}{\sqrt {\frac {1}{\log ^2(K[5])}}}dK[5]-1+c_1}{\sqrt {\frac {1}{\log ^2(x)}} \int _1^x\frac {F(K[5])}{\sqrt {\frac {1}{\log ^2(K[5])}}}dK[5]+c_1 \sqrt {\frac {1}{\log ^2(x)}}}\right \}\right \}\] ✓ Maple : cpu = 0.034 (sec), leaf count = 35
dsolve(diff(y(x),x) = -F(x)*(-y(x)^2-2*y(x)*ln(x)-ln(x)^2)+1/ln(x)/x*y(x),y(x))
\[y \left (x \right ) = -\frac {\ln \left (x \right ) \left (\int -2 F \left (x \right ) \ln \left (x \right )d x -c_{1}-2\right )}{\int -2 F \left (x \right ) \ln \left (x \right )d x -c_{1}}\]