\[ y'(x)=\frac {2 x F(y(x)+\log (2 x+1))+F(y(x)+\log (2 x+1))-2}{2 x+1} \] ✓ Mathematica : cpu = 20.9561 (sec), leaf count = 102
\[\text {Solve}\left [c_1=\int _1^{y(x)} \left (\frac {1}{F(K[2]+\log (2 x+1))}-\int _1^x -\frac {2 F'(K[2]+\log (2 K[1]+1))}{(2 K[1]+1) F(K[2]+\log (2 K[1]+1))^2} \, dK[1]\right ) \, dK[2]+\int _1^x \left (\frac {2}{(2 K[1]+1) F(\log (2 K[1]+1)+y(x))}-1\right ) \, dK[1],y(x)\right ]\]
✓ Maple : cpu = 0.157 (sec), leaf count = 27
\[ \left \{ y \left ( x \right ) =-\ln \left ( 2\,x+1 \right ) +{\it RootOf} \left ( -x+\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+{\it \_C1} \right ) \right \} \]