ODE
\[ y(x) (y(x)+1) y'(x)=x (x+1) \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.0157604 (sec), leaf count = 346
\[\left \{\left \{y(x)\to \frac {1}{2} \left (\sqrt [3]{\sqrt {\left (12 c_1+4 x^3+6 x^2-1\right ){}^2-1}+12 c_1+4 x^3+6 x^2-1}+\frac {1}{\sqrt [3]{\sqrt {\left (12 c_1+4 x^3+6 x^2-1\right ){}^2-1}+12 c_1+4 x^3+6 x^2-1}}-1\right )\right \},\left \{y(x)\to \frac {1}{8} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {\left (12 c_1+4 x^3+6 x^2-1\right ){}^2-1}+12 c_1+4 x^3+6 x^2-1}+\frac {-2-2 i \sqrt {3}}{\sqrt [3]{\sqrt {\left (12 c_1+4 x^3+6 x^2-1\right ){}^2-1}+12 c_1+4 x^3+6 x^2-1}}-4\right )\right \},\left \{y(x)\to \frac {1}{8} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (12 c_1+4 x^3+6 x^2-1\right ){}^2-1}+12 c_1+4 x^3+6 x^2-1}+\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{\sqrt {\left (12 c_1+4 x^3+6 x^2-1\right ){}^2-1}+12 c_1+4 x^3+6 x^2-1}}-4\right )\right \}\right \}\]
Maple ✓
cpu = 0.008 (sec), leaf count = 27
\[ \left \{ {\frac {{x}^{3}}{3}}+{\frac {{x}^{2}}{2}}-{\frac { \left ( y \left ( x \right ) \right ) ^{3}}{3}}-{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{2}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[y[x]*(1 + y[x])*y'[x] == x*(1 + x),y[x],x]
Mathematica raw output
{{y[x] -> (-1 + (-1 + 6*x^2 + 4*x^3 + 12*C[1] + Sqrt[-1 + (-1 + 6*x^2 + 4*x^3 + 12*C[1])^2])^(-1/3) + (-1 + 6*x^2 + 4*x^3 + 12*C[1] + Sqrt[-1 + (-1 + 6*x^2 + 4*x^3 + 12*C[1])^2])^(1/3))/2}, {y[x] -> (-4 + (-2 - (2*I)*Sqrt[3])/(-1 + 6*x^2 + 4*x^3 + 12*C[1] + Sqrt[-1 + (-1 + 6*x^2 + 4*x^3 + 12*C[1])^2])^(1/3) + (2*I)*(I + Sqrt[3])*(-1 + 6*x^2 + 4*x^3 + 12*C[1] + Sqrt[-1 + (-1 + 6*x^2 + 4*x^3 + 12*C[1])^2])^(1/3))/8}, {y[x] -> (-4 + ((2*I)*(I + Sqrt[3]))/(-1 + 6*x^2 + 4*x^3 + 12*C[1] + Sqrt[-1 + (-1 + 6*x^2 + 4*x^3 + 12*C[1])^2])^(1/3) - 2*(1 + I*Sqrt[3])*(-1 + 6*x^2 + 4*x^3 + 12*C[1] + Sqrt[-1 + (-1 + 6*x^2 + 4*x^3 + 12*C[1])^2])^(1/3))/8}}
Maple raw input
dsolve(y(x)*(1+y(x))*diff(y(x),x) = (1+x)*x, y(x),'implicit')
Maple raw output
1/3*x^3+1/2*x^2-1/3*y(x)^3-1/2*y(x)^2+_C1 = 0