\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {-2\,y \left ( x \right ) -2\,\ln \left ( 2\,x+1 \right ) -2+2\,x \left ( y \left ( x \right ) \right ) ^{3}+ \left ( y \left ( x \right ) \right ) ^{3}+6\, \left ( y \left ( x \right ) \right ) ^{2}\ln \left ( 2\,x+1 \right ) x+3\, \left ( y \left ( x \right ) \right ) ^{2}\ln \left ( 2\,x+1 \right ) +6\,y \left ( x \right ) \left ( \ln \left ( 2\,x+1 \right ) \right ) ^{2}x+3\,y \left ( x \right ) \left ( \ln \left ( 2\,x+1 \right ) \right ) ^{2}+2\, \left ( \ln \left ( 2\,x+1 \right ) \right ) ^{3}x+ \left ( \ln \left ( 2\,x+1 \right ) \right ) ^{3}}{ \left ( 2\,x+1 \right ) \left ( y \left ( x \right ) +\ln \left ( 2\,x+1 \right ) +1 \right ) }}=0} \]
Mathematica: cpu = 0.054007 (sec), leaf count = 124 \[ \left \{\left \{y(x)\to \frac {1}{(2 x+1) \left (\frac {2 x+1}{4 x^2+4 x+1}-\frac {1}{(2 x+1) \sqrt {c_1-2 x}}\right )}-\log (2 x+1)-1\right \},\left \{y(x)\to \frac {1}{(2 x+1) \left (\frac {1}{(2 x+1) \sqrt {c_1-2 x}}+\frac {2 x+1}{4 x^2+4 x+1}\right )}-\log (2 x+1)-1\right \}\right \} \]
Maple: cpu = 0.046 (sec), leaf count = 79 \[ \left \{ y \left ( x \right ) =-{1 \left ( \sqrt {{\it \_C1}-2\,x}\ln \left ( 2\,x+1 \right ) -\ln \left ( 2\,x+1 \right ) -1 \right ) \left ( \sqrt {{\it \_C1}-2\,x}-1 \right ) ^{-1}},y \left ( x \right ) =-{1 \left ( \sqrt {{\it \_C1}-2\,x}\ln \left ( 2\,x+1 \right ) +\ln \left ( 2\,x+1 \right ) +1 \right ) \left ( \sqrt {{\it \_C1}-2\,x}+1 \right ) ^{-1}} \right \} \]