\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {2\,F \left ( y \left ( x \right ) +\ln \left ( 2\,x+1 \right ) \right ) x+F \left ( y \left ( x \right ) +\ln \left ( 2\,x+1 \right ) \right ) -2}{2\,x+1}}=0} \]
Mathematica: cpu = 17.469218 (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 = 0.124 (sec), leaf count = 27 \[ \left \{ y \left ( x \right ) =-\ln \left ( 2\,x+1 \right ) +{\it RootOf} \left ( -x+\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+{\it \_C1} \right ) \right \} \]