##### 4.14.9 $$x^3+3 x \left (y(x)^2+x\right ) y'(x)-3 x y(x)-2 y(x)^3=0$$

ODE
$x^3+3 x \left (y(x)^2+x\right ) y'(x)-3 x y(x)-2 y(x)^3=0$ ODE Classiﬁcation

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.378553 (sec), leaf count = 362

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

Maple
cpu = 0.043 (sec), leaf count = 450

$\left [y \left (x \right ) = \frac {\left (-4 x^{2} \textit {\_C1} -4 x^{3}+4 \sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1} \,x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{2}-\frac {2 x}{\left (-4 x^{2} \textit {\_C1} -4 x^{3}+4 \sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1} \,x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (-4 x^{2} \textit {\_C1} -4 x^{3}+4 \sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1} \,x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (-4 x^{2} \textit {\_C1} -4 x^{3}+4 \sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1} \,x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-4 x^{2} \textit {\_C1} -4 x^{3}+4 \sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1} \,x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (-4 x^{2} \textit {\_C1} -4 x^{3}+4 \sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1} \,x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (-4 x^{2} \textit {\_C1} -4 x^{3}+4 \sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1} \,x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{4}+\frac {x}{\left (-4 x^{2} \textit {\_C1} -4 x^{3}+4 \sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1} \,x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-4 x^{2} \textit {\_C1} -4 x^{3}+4 \sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1} \,x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}{2}+\frac {2 x}{\left (-4 x^{2} \textit {\_C1} -4 x^{3}+4 \sqrt {x^{4} \textit {\_C1}^{2}+2 \textit {\_C1} \,x^{5}+x^{6}+4 x^{3}}\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

DSolve[x^3 - 3*x*y[x] - 2*y[x]^3 + 3*x*(x + y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((2^(1/3)*x)/(-x^3 + x^2*C[1] + Sqrt[x^3*(4 + x^3 - 2*x^2*C[1] + x*C[
1]^2)])^(1/3)) + (-x^3 + x^2*C[1] + Sqrt[x^3*(4 + x^3 - 2*x^2*C[1] + x*C[1]^2)])
^(1/3)/2^(1/3)}, {y[x] -> (2^(1/3)*(2 + (2*I)*Sqrt[3])*x + I*2^(2/3)*(I + Sqrt[3
])*(-x^3 + x^2*C[1] + Sqrt[x^3*(4 + x^3 - 2*x^2*C[1] + x*C[1]^2)])^(2/3))/(4*(-x
^3 + x^2*C[1] + Sqrt[x^3*(4 + x^3 - 2*x^2*C[1] + x*C[1]^2)])^(1/3))}, {y[x] -> (
2^(1/3)*(2 - (2*I)*Sqrt[3])*x - I*2^(2/3)*(-I + Sqrt[3])*(-x^3 + x^2*C[1] + Sqrt
[x^3*(4 + x^3 - 2*x^2*C[1] + x*C[1]^2)])^(2/3))/(4*(-x^3 + x^2*C[1] + Sqrt[x^3*(
4 + x^3 - 2*x^2*C[1] + x*C[1]^2)])^(1/3))}}

Maple raw input

dsolve(3*x*(x+y(x)^2)*diff(y(x),x)+x^3-3*x*y(x)-2*y(x)^3 = 0, y(x))

Maple raw output

[y(x) = 1/2*(-4*x^2*_C1-4*x^3+4*(_C1^2*x^4+2*_C1*x^5+x^6+4*x^3)^(1/2))^(1/3)-2*x
/(-4*x^2*_C1-4*x^3+4*(_C1^2*x^4+2*_C1*x^5+x^6+4*x^3)^(1/2))^(1/3), y(x) = -1/4*(
-4*x^2*_C1-4*x^3+4*(_C1^2*x^4+2*_C1*x^5+x^6+4*x^3)^(1/2))^(1/3)+x/(-4*x^2*_C1-4*
x^3+4*(_C1^2*x^4+2*_C1*x^5+x^6+4*x^3)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-4*x^2*_C
1-4*x^3+4*(_C1^2*x^4+2*_C1*x^5+x^6+4*x^3)^(1/2))^(1/3)+2*x/(-4*x^2*_C1-4*x^3+4*(
_C1^2*x^4+2*_C1*x^5+x^6+4*x^3)^(1/2))^(1/3)), y(x) = -1/4*(-4*x^2*_C1-4*x^3+4*(_
C1^2*x^4+2*_C1*x^5+x^6+4*x^3)^(1/2))^(1/3)+x/(-4*x^2*_C1-4*x^3+4*(_C1^2*x^4+2*_C
1*x^5+x^6+4*x^3)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-4*x^2*_C1-4*x^3+4*(_C1^2*x^4+
2*_C1*x^5+x^6+4*x^3)^(1/2))^(1/3)+2*x/(-4*x^2*_C1-4*x^3+4*(_C1^2*x^4+2*_C1*x^5+x
^6+4*x^3)^(1/2))^(1/3))]