##### 4.14.46 $$x \left (2 x^3+y(x)^3\right ) y'(x)=y(x) \left (2 x^3-x^2 y(x)+y(x)^3\right )$$

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

[[_homogeneous, class A], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.449021 (sec), leaf count = 362

$\left \{\left \{y(x)\to \frac {-6^{2/3} x^2 \log (x)+6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}\right ){}^{2/3}}{3 \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}\right \},\left \{y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}{6^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) x^2 (\log (x)-c_1)}{\sqrt [3]{6} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}\right \},\left \{y(x)\to \frac {i \left (\sqrt {3}+i\right ) x^2 (-\log (x)+c_1)}{\sqrt [3]{6} \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{9 x^3+\sqrt {3} \sqrt {x^6 \left (27+2 (\log (x)-c_1){}^3\right )}}}{6^{2/3}}\right \}\right \}$

Maple
cpu = 0.046 (sec), leaf count = 443

$\left [y \left (x \right ) = \left (\frac {\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} \textit {\_C1} +18 \ln \left (x \right ) \textit {\_C1}^{2}+6 \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}{3}-\frac {3 \left (\frac {2 \ln \left (x \right )}{3}+\frac {2 \textit {\_C1}}{3}\right )}{\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} \textit {\_C1} +18 \ln \left (x \right ) \textit {\_C1}^{2}+6 \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}\right ) x, y \left (x \right ) = \left (-\frac {\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} \textit {\_C1} +18 \ln \left (x \right ) \textit {\_C1}^{2}+6 \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\frac {\ln \left (x \right )+\textit {\_C1}}{\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} \textit {\_C1} +18 \ln \left (x \right ) \textit {\_C1}^{2}+6 \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} \textit {\_C1} +18 \ln \left (x \right ) \textit {\_C1}^{2}+6 \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}{3}+\frac {2 \ln \left (x \right )+2 \textit {\_C1}}{\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} \textit {\_C1} +18 \ln \left (x \right ) \textit {\_C1}^{2}+6 \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}\right ) x, y \left (x \right ) = \left (-\frac {\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} \textit {\_C1} +18 \ln \left (x \right ) \textit {\_C1}^{2}+6 \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\frac {\ln \left (x \right )+\textit {\_C1}}{\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} \textit {\_C1} +18 \ln \left (x \right ) \textit {\_C1}^{2}+6 \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} \textit {\_C1} +18 \ln \left (x \right ) \textit {\_C1}^{2}+6 \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}{3}+\frac {2 \ln \left (x \right )+2 \textit {\_C1}}{\left (54+6 \sqrt {6 \ln \left (x \right )^{3}+18 \ln \left (x \right )^{2} \textit {\_C1} +18 \ln \left (x \right ) \textit {\_C1}^{2}+6 \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}\right ) x\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (6^(2/3)*x^2*C[1] - 6^(2/3)*x^2*Log[x] + 6^(1/3)*(9*x^3 + Sqrt[3]*Sqrt
[x^6*(27 + 2*(-C[1] + Log[x])^3)])^(2/3))/(3*(9*x^3 + Sqrt[3]*Sqrt[x^6*(27 + 2*(
-C[1] + Log[x])^3)])^(1/3))}, {y[x] -> ((1 + I*Sqrt[3])*x^2*(-C[1] + Log[x]))/(6
^(1/3)*(9*x^3 + Sqrt[3]*Sqrt[x^6*(27 + 2*(-C[1] + Log[x])^3)])^(1/3)) + (I*(I +
Sqrt[3])*(9*x^3 + Sqrt[3]*Sqrt[x^6*(27 + 2*(-C[1] + Log[x])^3)])^(1/3))/6^(2/3)}
, {y[x] -> (I*(I + Sqrt[3])*x^2*(C[1] - Log[x]))/(6^(1/3)*(9*x^3 + Sqrt[3]*Sqrt[
x^6*(27 + 2*(-C[1] + Log[x])^3)])^(1/3)) - ((1 + I*Sqrt[3])*(9*x^3 + Sqrt[3]*Sqr
t[x^6*(27 + 2*(-C[1] + Log[x])^3)])^(1/3))/6^(2/3)}}

Maple raw input

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

Maple raw output

[y(x) = (1/3*(54+6*(6*ln(x)^3+18*ln(x)^2*_C1+18*ln(x)*_C1^2+6*_C1^3+81)^(1/2))^(
1/3)-3*(2/3*ln(x)+2/3*_C1)/(54+6*(6*ln(x)^3+18*ln(x)^2*_C1+18*ln(x)*_C1^2+6*_C1^
3+81)^(1/2))^(1/3))*x, y(x) = (-1/6*(54+6*(6*ln(x)^3+18*ln(x)^2*_C1+18*ln(x)*_C1
^2+6*_C1^3+81)^(1/2))^(1/3)+3/2*(2/3*ln(x)+2/3*_C1)/(54+6*(6*ln(x)^3+18*ln(x)^2*
_C1+18*ln(x)*_C1^2+6*_C1^3+81)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3*(54+6*(6*ln(x)^3+
18*ln(x)^2*_C1+18*ln(x)*_C1^2+6*_C1^3+81)^(1/2))^(1/3)+3*(2/3*ln(x)+2/3*_C1)/(54
+6*(6*ln(x)^3+18*ln(x)^2*_C1+18*ln(x)*_C1^2+6*_C1^3+81)^(1/2))^(1/3)))*x, y(x) =
(-1/6*(54+6*(6*ln(x)^3+18*ln(x)^2*_C1+18*ln(x)*_C1^2+6*_C1^3+81)^(1/2))^(1/3)+3
/2*(2/3*ln(x)+2/3*_C1)/(54+6*(6*ln(x)^3+18*ln(x)^2*_C1+18*ln(x)*_C1^2+6*_C1^3+81
)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3*(54+6*(6*ln(x)^3+18*ln(x)^2*_C1+18*ln(x)*_C1^2
+6*_C1^3+81)^(1/2))^(1/3)+3*(2/3*ln(x)+2/3*_C1)/(54+6*(6*ln(x)^3+18*ln(x)^2*_C1+
18*ln(x)*_C1^2+6*_C1^3+81)^(1/2))^(1/3)))*x]