\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {2\,x \left ( y \left ( x \right ) \right ) ^{2}+4\,y \left ( x \right ) \ln \left ( 2\,x+1 \right ) x+2\, \left ( \ln \left ( 2\,x+1 \right ) \right ) ^{2}x+ \left ( y \left ( x \right ) \right ) ^{2}-2+ \left ( \ln \left ( 2\,x+1 \right ) \right ) ^{2}+2\,y \left ( x \right ) \ln \left ( 2\,x+1 \right ) }{2\,x+1}}=0} \]
Mathematica: cpu = 0.022503 (sec), leaf count = 22 \[ \left \{\left \{y(x)\to \frac {1}{c_1-x}-\log (2 x+1)\right \}\right \} \]
Maple: cpu = 0.062 (sec), leaf count = 66 \[ \left \{ y \left ( x \right ) =-{\frac {1}{2} \left ( 4\,{\frac {\ln \left ( 2\,x+1 \right ) x}{2\,x+1}}+2\,{\frac {\ln \left ( 2\,x+1 \right ) }{2\,x+1}} \right ) \left ( 2\,{\frac {x}{2\,x+1}}+ \left ( 2\, x+1 \right ) ^{-1} \right ) ^{-1}}+ \left ( {\it \_C1}-x \right ) ^{-1} \right \} \]