\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac {{x}^{3} \left ( 3\,x+3+\sqrt {9\,{x}^{4}-4\, \left ( y \left ( x \right ) \right ) ^{3}} \right ) }{ \left ( 1+x \right ) \left ( y \left ( x \right ) \right ) ^{2}}}=0} \]
Mathematica: cpu = 6.127778 (sec), leaf count = 314 \[ \left \{\left \{y(x)\to \sqrt [3]{6 c_1 x^3-9 c_1 x^2+18 c_1 x-18 c_1 \log (x+1)-9 c_1^2-x^6+3 x^5-6 x^4+9 x^3+6 x^3 \log (x+1)-9 x^2-9 x^2 \log (x+1)-9 \log ^2(x+1)+18 x \log (x+1)}\right \},\left \{y(x)\to -\sqrt [3]{-1} \sqrt [3]{6 c_1 x^3-9 c_1 x^2+18 c_1 x-18 c_1 \log (x+1)-9 c_1^2-x^6+3 x^5-6 x^4+9 x^3+6 x^3 \log (x+1)-9 x^2-9 x^2 \log (x+1)-9 \log ^2(x+1)+18 x \log (x+1)}\right \},\left \{y(x)\to (-1)^{2/3} \sqrt [3]{6 c_1 x^3-9 c_1 x^2+18 c_1 x-18 c_1 \log (x+1)-9 c_1^2-x^6+3 x^5-6 x^4+9 x^3+6 x^3 \log (x+1)-9 x^2-9 x^2 \log (x+1)-9 \log ^2(x+1)+18 x \log (x+1)}\right \}\right \} \]
Maple: cpu = 0.171 (sec), leaf count = 48 \[ \left \{ \int _{{\it \_b}}^{y \left ( x \right ) }\!{{{\it \_a}}^{2}{ \frac {1}{\sqrt {9\,{x}^{4}-4\,{{\it \_a}}^{3}}}}}\,{\rm d}{\it \_a}-{ \frac {{x}^{3}}{3}}+{\frac {{x}^{2}}{2}}-x+\ln \left ( 1+x \right ) -{ \it \_C1}=0 \right \} \]