\[ y'(x)=\frac {2 x F(y(x)+\log (2 x+1))+F(y(x)+\log (2 x+1))-2}{2 x+1} \] ✓ Mathematica : cpu = 20.0878 (sec), leaf count = 114
\[\text {Solve}\left [\int _1^{y(x)} -\frac {F(K[2]+\log (2 x+1)) \int _1^x -\frac {2 F'(K[2]+\log (2 K[1]+1))}{(2 K[1]+1) F(K[2]+\log (2 K[1]+1))^2} \, dK[1]-1}{F(K[2]+\log (2 x+1))} \, dK[2]+\int _1^x \left (\frac {2}{(2 K[1]+1) F(\log (2 K[1]+1)+y(x))}-1\right ) \, dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.153 (sec), leaf count = 27
\[ \left \{ y \left ( x \right ) =-\ln \left ( 2\,x+1 \right ) +{\it RootOf} \left ( -x+\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+{\it \_C1} \right ) \right \} \]