\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac { \left ( x+y \left ( x \right ) +1 \right ) y \left ( x \right ) }{ \left ( \left ( y \left ( x \right ) \right ) ^{4}+ \left ( y \left ( x \right ) \right ) ^{3}+ \left ( y \left ( x \right ) \right ) ^{2}+x \right ) \left ( 1+x \right ) }}=0} \]
Mathematica: cpu = 0.580574 (sec), leaf count = 2497 \[ \left \{\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}+\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}-\frac {1}{2} \sqrt {-\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}-\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}-\frac {24 \left (c_1+\log (x+1)\right )+\frac {117}{8}}{4 \sqrt {\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}+\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}}-\frac {23}{8}}-\frac {3}{8}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}+\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}+\frac {1}{2} \sqrt {-\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}-\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}-\frac {24 \left (c_1+\log (x+1)\right )+\frac {117}{8}}{4 \sqrt {\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}+\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}}-\frac {23}{8}}-\frac {3}{8}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}+\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}-\frac {1}{2} \sqrt {-\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}-\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}+\frac {24 \left (c_1+\log (x+1)\right )+\frac {117}{8}}{4 \sqrt {\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}+\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}}-\frac {23}{8}}-\frac {3}{8}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}+\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}+\frac {1}{2} \sqrt {-\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}-\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}+\frac {24 \left (c_1+\log (x+1)\right )+\frac {117}{8}}{4 \sqrt {\frac {3 \sqrt [3]{2} \left (-8 x+3 c_1+3 \log (x+1)+2\right )}{\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}+\frac {\sqrt [3]{1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+\sqrt {\left (1944 \left (c_1+\log (x+1)\right ){}^2+972 \left (c_1+\log (x+1)\right )+3726 x+432\right ){}^2-4 \left (-144 x+54 \left (c_1+\log (x+1)\right )+36\right ){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}}-\frac {23}{8}}-\frac {3}{8}\right \}\right \} \]
Maple: cpu = 0.125 (sec), leaf count = 31 \[ \left \{ \ln \left ( 1+x \right ) +{\frac {x}{y \left ( x \right ) }}-{ \frac { \left ( y \left ( x \right ) \right ) ^{3}}{3}}-{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{2}}-y \left ( x \right ) +{\it \_C1}=0 \right \} \]