\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) =-F \left ( x \right ) \left ( - \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) \ln \left ( x \right ) - \left ( \ln \left ( x \right ) \right ) ^{2} \right ) +{\frac {y \left ( x \right ) }{x\ln \left ( x \right ) }}=0} \]
Mathematica: cpu = 5232.495943 (sec), leaf count = 71 \[ \left \{\left \{y(x)\to \frac {\int _1^x \frac {F(K[5])}{\sqrt {\frac {1}{\log ^2(K[5])}}} \, dK[5]+c_1-1}{\sqrt {\frac {1}{\log ^2(x)}} \left (\int _1^x \frac {F(K[5])}{\sqrt {\frac {1}{\log ^2(K[5])}}} \, dK[5]\right )+c_1 \sqrt {\frac {1}{\log ^2(x)}}}\right \}\right \} \]
Maple: cpu = 0.016 (sec), leaf count = 35 \[ \left \{ y \left ( x \right ) =-{\frac {\ln \left ( x \right ) \left ( \int \!-2\,\ln \left ( x \right ) F \left ( x \right ) \,{\rm d}x-{\it \_C1}-2 \right ) }{\int \!-2\,\ln \left ( x \right ) F \left ( x \right ) \,{\rm d}x-{\it \_C1}}} \right \} \]