4.14.8 $$x \left (x-3 y(x)^2\right ) y'(x)+y(x) \left (2 x-y(x)^2\right )=0$$

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

[[_homogeneous, class G], _exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.367367 (sec), leaf count = 328

$\left \{\left \{y(x)\to -\frac {2 \sqrt [3]{3} x^3+\sqrt [3]{2} \left (9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{6^{2/3} x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}}\right \},\left \{y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^3+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}}\right \},\left \{y(x)\to \frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^3+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}}\right \}\right \}$

Maple
cpu = 0.235 (sec), leaf count = 327

$\left [y \left (x \right ) = \frac {\left (\left (12 \sqrt {-12 x^{5}+81 \textit {\_C1}^{2}}+108 \textit {\_C1} \right ) x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {2 x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 \textit {\_C1}^{2}}+108 \textit {\_C1} \right ) x^{2}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (\left (12 \sqrt {-12 x^{5}+81 \textit {\_C1}^{2}}+108 \textit {\_C1} \right ) x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 \textit {\_C1}^{2}}+108 \textit {\_C1} \right ) x^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (12 \sqrt {-12 x^{5}+81 \textit {\_C1}^{2}}+108 \textit {\_C1} \right ) x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {2 x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 \textit {\_C1}^{2}}+108 \textit {\_C1} \right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (\left (12 \sqrt {-12 x^{5}+81 \textit {\_C1}^{2}}+108 \textit {\_C1} \right ) x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 \textit {\_C1}^{2}}+108 \textit {\_C1} \right ) x^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (12 \sqrt {-12 x^{5}+81 \textit {\_C1}^{2}}+108 \textit {\_C1} \right ) x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {2 x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 \textit {\_C1}^{2}}+108 \textit {\_C1} \right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -((2*3^(1/3)*x^3 + 2^(1/3)*(9*x^2*C[1] + Sqrt[-12*x^9 + 81*x^4*C[1]^2]
)^(2/3))/(6^(2/3)*x*(9*x^2*C[1] + Sqrt[-12*x^9 + 81*x^4*C[1]^2])^(1/3)))}, {y[x]
 -> (2*2^(1/3)*3^(1/6)*(3*I + Sqrt[3])*x^3 + 3^(1/3)*(1 - I*Sqrt[3])*(18*x^2*C[1
] + 2*Sqrt[-12*x^9 + 81*x^4*C[1]^2])^(2/3))/(12*x*(9*x^2*C[1] + Sqrt[-12*x^9 + 8
1*x^4*C[1]^2])^(1/3))}, {y[x] -> (2*2^(1/3)*3^(1/6)*(-3*I + Sqrt[3])*x^3 + 3^(1/
3)*(1 + I*Sqrt[3])*(18*x^2*C[1] + 2*Sqrt[-12*x^9 + 81*x^4*C[1]^2])^(2/3))/(12*x*
(9*x^2*C[1] + Sqrt[-12*x^9 + 81*x^4*C[1]^2])^(1/3))}}

Maple raw input

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

Maple raw output

[y(x) = 1/6/x*((12*(-12*x^5+81*_C1^2)^(1/2)+108*_C1)*x^2)^(1/3)+2*x^2/((12*(-12*
x^5+81*_C1^2)^(1/2)+108*_C1)*x^2)^(1/3), y(x) = -1/12/x*((12*(-12*x^5+81*_C1^2)^
(1/2)+108*_C1)*x^2)^(1/3)-x^2/((12*(-12*x^5+81*_C1^2)^(1/2)+108*_C1)*x^2)^(1/3)-
1/2*I*3^(1/2)*(1/6/x*((12*(-12*x^5+81*_C1^2)^(1/2)+108*_C1)*x^2)^(1/3)-2*x^2/((1
2*(-12*x^5+81*_C1^2)^(1/2)+108*_C1)*x^2)^(1/3)), y(x) = -1/12/x*((12*(-12*x^5+81
*_C1^2)^(1/2)+108*_C1)*x^2)^(1/3)-x^2/((12*(-12*x^5+81*_C1^2)^(1/2)+108*_C1)*x^2
)^(1/3)+1/2*I*3^(1/2)*(1/6/x*((12*(-12*x^5+81*_C1^2)^(1/2)+108*_C1)*x^2)^(1/3)-2
*x^2/((12*(-12*x^5+81*_C1^2)^(1/2)+108*_C1)*x^2)^(1/3))]