##### 4.13.12 $$y(x) (y(x)+1) y'(x)=x (x+1)$$

ODE
$y(x) (y(x)+1) y'(x)=x (x+1)$ ODE Classiﬁcation

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.324303 (sec), leaf count = 346

$\left \{\left \{y(x)\to \frac {1}{2} \left (\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {1}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-1\right )\right \},\left \{y(x)\to \frac {1}{8} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {-2-2 i \sqrt {3}}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-4\right )\right \},\left \{y(x)\to \frac {1}{8} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}+\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{4 x^3+6 x^2+\sqrt {-1+\left (4 x^3+6 x^2-1+12 c_1\right ){}^2}-1+12 c_1}}-4\right )\right \}\right \}$

Maple
cpu = 0.031 (sec), leaf count = 720

$\left [y \left (x \right ) = \frac {\left (-1+4 x^{3}+6 x^{2}+12 \textit {\_C1} +2 \sqrt {4 x^{6}+12 x^{5}+24 \textit {\_C1} \,x^{3}+9 x^{4}+36 x^{2} \textit {\_C1} -2 x^{3}+36 \textit {\_C1}^{2}-3 x^{2}-6 \textit {\_C1}}\right )^{\frac {1}{3}}}{2}+\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 \textit {\_C1} +2 \sqrt {4 x^{6}+12 x^{5}+24 \textit {\_C1} \,x^{3}+9 x^{4}+36 x^{2} \textit {\_C1} -2 x^{3}+36 \textit {\_C1}^{2}-3 x^{2}-6 \textit {\_C1}}\right )^{\frac {1}{3}}}-\frac {1}{2}, y \left (x \right ) = -\frac {\left (-1+4 x^{3}+6 x^{2}+12 \textit {\_C1} +2 \sqrt {4 x^{6}+12 x^{5}+24 \textit {\_C1} \,x^{3}+9 x^{4}+36 x^{2} \textit {\_C1} -2 x^{3}+36 \textit {\_C1}^{2}-3 x^{2}-6 \textit {\_C1}}\right )^{\frac {1}{3}}}{4}-\frac {1}{4 \left (-1+4 x^{3}+6 x^{2}+12 \textit {\_C1} +2 \sqrt {4 x^{6}+12 x^{5}+24 \textit {\_C1} \,x^{3}+9 x^{4}+36 x^{2} \textit {\_C1} -2 x^{3}+36 \textit {\_C1}^{2}-3 x^{2}-6 \textit {\_C1}}\right )^{\frac {1}{3}}}-\frac {1}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-1+4 x^{3}+6 x^{2}+12 \textit {\_C1} +2 \sqrt {4 x^{6}+12 x^{5}+24 \textit {\_C1} \,x^{3}+9 x^{4}+36 x^{2} \textit {\_C1} -2 x^{3}+36 \textit {\_C1}^{2}-3 x^{2}-6 \textit {\_C1}}\right )^{\frac {1}{3}}}{2}-\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 \textit {\_C1} +2 \sqrt {4 x^{6}+12 x^{5}+24 \textit {\_C1} \,x^{3}+9 x^{4}+36 x^{2} \textit {\_C1} -2 x^{3}+36 \textit {\_C1}^{2}-3 x^{2}-6 \textit {\_C1}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (-1+4 x^{3}+6 x^{2}+12 \textit {\_C1} +2 \sqrt {4 x^{6}+12 x^{5}+24 \textit {\_C1} \,x^{3}+9 x^{4}+36 x^{2} \textit {\_C1} -2 x^{3}+36 \textit {\_C1}^{2}-3 x^{2}-6 \textit {\_C1}}\right )^{\frac {1}{3}}}{4}-\frac {1}{4 \left (-1+4 x^{3}+6 x^{2}+12 \textit {\_C1} +2 \sqrt {4 x^{6}+12 x^{5}+24 \textit {\_C1} \,x^{3}+9 x^{4}+36 x^{2} \textit {\_C1} -2 x^{3}+36 \textit {\_C1}^{2}-3 x^{2}-6 \textit {\_C1}}\right )^{\frac {1}{3}}}-\frac {1}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-1+4 x^{3}+6 x^{2}+12 \textit {\_C1} +2 \sqrt {4 x^{6}+12 x^{5}+24 \textit {\_C1} \,x^{3}+9 x^{4}+36 x^{2} \textit {\_C1} -2 x^{3}+36 \textit {\_C1}^{2}-3 x^{2}-6 \textit {\_C1}}\right )^{\frac {1}{3}}}{2}-\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 \textit {\_C1} +2 \sqrt {4 x^{6}+12 x^{5}+24 \textit {\_C1} \,x^{3}+9 x^{4}+36 x^{2} \textit {\_C1} -2 x^{3}+36 \textit {\_C1}^{2}-3 x^{2}-6 \textit {\_C1}}\right )^{\frac {1}{3}}}\right )}{2}\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) = (x+1)*x, y(x))

Maple raw output

[y(x) = 1/2*(-1+4*x^3+6*x^2+12*_C1+2*(4*x^6+12*x^5+24*_C1*x^3+9*x^4+36*_C1*x^2-2
*x^3+36*_C1^2-3*x^2-6*_C1)^(1/2))^(1/3)+1/2/(-1+4*x^3+6*x^2+12*_C1+2*(4*x^6+12*x
^5+24*_C1*x^3+9*x^4+36*_C1*x^2-2*x^3+36*_C1^2-3*x^2-6*_C1)^(1/2))^(1/3)-1/2, y(x
) = -1/4*(-1+4*x^3+6*x^2+12*_C1+2*(4*x^6+12*x^5+24*_C1*x^3+9*x^4+36*_C1*x^2-2*x^
3+36*_C1^2-3*x^2-6*_C1)^(1/2))^(1/3)-1/4/(-1+4*x^3+6*x^2+12*_C1+2*(4*x^6+12*x^5+
24*_C1*x^3+9*x^4+36*_C1*x^2-2*x^3+36*_C1^2-3*x^2-6*_C1)^(1/2))^(1/3)-1/2-1/2*I*3
^(1/2)*(1/2*(-1+4*x^3+6*x^2+12*_C1+2*(4*x^6+12*x^5+24*_C1*x^3+9*x^4+36*_C1*x^2-2
*x^3+36*_C1^2-3*x^2-6*_C1)^(1/2))^(1/3)-1/2/(-1+4*x^3+6*x^2+12*_C1+2*(4*x^6+12*x
^5+24*_C1*x^3+9*x^4+36*_C1*x^2-2*x^3+36*_C1^2-3*x^2-6*_C1)^(1/2))^(1/3)), y(x) =
 -1/4*(-1+4*x^3+6*x^2+12*_C1+2*(4*x^6+12*x^5+24*_C1*x^3+9*x^4+36*_C1*x^2-2*x^3+3
6*_C1^2-3*x^2-6*_C1)^(1/2))^(1/3)-1/4/(-1+4*x^3+6*x^2+12*_C1+2*(4*x^6+12*x^5+24*
_C1*x^3+9*x^4+36*_C1*x^2-2*x^3+36*_C1^2-3*x^2-6*_C1)^(1/2))^(1/3)-1/2+1/2*I*3^(1
/2)*(1/2*(-1+4*x^3+6*x^2+12*_C1+2*(4*x^6+12*x^5+24*_C1*x^3+9*x^4+36*_C1*x^2-2*x^
3+36*_C1^2-3*x^2-6*_C1)^(1/2))^(1/3)-1/2/(-1+4*x^3+6*x^2+12*_C1+2*(4*x^6+12*x^5+
24*_C1*x^3+9*x^4+36*_C1*x^2-2*x^3+36*_C1^2-3*x^2-6*_C1)^(1/2))^(1/3))]