\[ y'(x)=\frac {2 x y(x)^3+y(x)^3-2 y(x)+6 x y(x) \log ^2(2 x+1)+3 y(x) \log ^2(2 x+1)+6 x y(x)^2 \log (2 x+1)+3 y(x)^2 \log (2 x+1)+2 x \log ^3(2 x+1)+\log ^3(2 x+1)-2 \log (2 x+1)-2}{(2 x+1) (y(x)+\log (2 x+1)+1)} \] ✓ Mathematica : cpu = 0.279504 (sec), leaf count = 124
DSolve[Derivative[1][y][x] == (-2 - 2*Log[1 + 2*x] + Log[1 + 2*x]^3 + 2*x*Log[1 + 2*x]^3 - 2*y[x] + 3*Log[1 + 2*x]^2*y[x] + 6*x*Log[1 + 2*x]^2*y[x] + 3*Log[1 + 2*x]*y[x]^2 + 6*x*Log[1 + 2*x]*y[x]^2 + y[x]^3 + 2*x*y[x]^3)/((1 + 2*x)*(1 + Log[1 + 2*x] + y[x])),y[x],x]
\[\left \{\left \{y(x)\to \frac {1}{(2 x+1) \left (\frac {2 x+1}{4 x^2+4 x+1}-\frac {1}{(2 x+1) \sqrt {-2 x+c_1}}\right )}-\log (2 x+1)-1\right \},\left \{y(x)\to \frac {1}{(2 x+1) \left (\frac {2 x+1}{4 x^2+4 x+1}+\frac {1}{(2 x+1) \sqrt {-2 x+c_1}}\right )}-\log (2 x+1)-1\right \}\right \}\] ✓ Maple : cpu = 0.08 (sec), leaf count = 79
dsolve(diff(y(x),x) = 1/(2*x+1)*(-2*y(x)-2*ln(2*x+1)-2+2*x*y(x)^3+y(x)^3+6*y(x)^2*ln(2*x+1)*x+3*y(x)^2*ln(2*x+1)+6*y(x)*ln(2*x+1)^2*x+3*y(x)*ln(2*x+1)^2+2*ln(2*x+1)^3*x+ln(2*x+1)^3)/(y(x)+ln(2*x+1)+1),y(x))
\[y \left (x \right ) = \frac {-\sqrt {c_{1}-2 x}\, \ln \left (2 x +1\right )+\ln \left (2 x +1\right )+1}{\sqrt {c_{1}-2 x}-1}\]