\[ y'(x)=\frac {y(x) (y(x)+x+1)}{(x+1) \left (y(x)^4+y(x)^3+y(x)^2+x\right )} \] ✓ Mathematica : cpu = 1.26496 (sec), leaf count = 2497
DSolve[Derivative[1][y][x] == (y[x]*(1 + x + y[x]))/((1 + x)*(x + y[x]^2 + y[x]^3 + y[x]^4)),y[x],x]
\[\left \{\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {3 \sqrt [3]{2} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}+\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}-\frac {1}{2} \sqrt {-\frac {3 \sqrt [3]{2} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}-\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}{6 \sqrt [3]{2}}-\frac {24 (c_1+\log (x+1))+\frac {117}{8}}{4 \sqrt {\frac {3 \sqrt [3]{2} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}+\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^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} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}+\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}+\frac {1}{2} \sqrt {-\frac {3 \sqrt [3]{2} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}-\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}{6 \sqrt [3]{2}}-\frac {24 (c_1+\log (x+1))+\frac {117}{8}}{4 \sqrt {\frac {3 \sqrt [3]{2} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}+\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^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} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}+\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}-\frac {1}{2} \sqrt {-\frac {3 \sqrt [3]{2} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}-\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}{6 \sqrt [3]{2}}+\frac {24 (c_1+\log (x+1))+\frac {117}{8}}{4 \sqrt {\frac {3 \sqrt [3]{2} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}+\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^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} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}+\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}+\frac {1}{2} \sqrt {-\frac {3 \sqrt [3]{2} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}-\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}{6 \sqrt [3]{2}}+\frac {24 (c_1+\log (x+1))+\frac {117}{8}}{4 \sqrt {\frac {3 \sqrt [3]{2} (-8 x+3 c_1+3 \log (x+1)+2)}{\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}+\frac {\sqrt [3]{1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+\sqrt {\left (1944 (c_1+\log (x+1)){}^2+972 (c_1+\log (x+1))+3726 x+432\right ){}^2-4 (-144 x+54 (c_1+\log (x+1))+36){}^3}+432}}{6 \sqrt [3]{2}}-\frac {23}{16}}}-\frac {23}{8}}-\frac {3}{8}\right \}\right \}\] ✓ Maple : cpu = 0.135 (sec), leaf count = 31
dsolve(diff(y(x),x) = 1/(y(x)^4+y(x)^3+y(x)^2+x)*(x+y(x)+1)*y(x)/(1+x),y(x))
\[\ln \left (1+x \right )+\frac {x}{y \left (x \right )}-\frac {y \left (x \right )^{3}}{3}-\frac {y \left (x \right )^{2}}{2}-y \left (x \right )+c_{1} = 0\]