##### 4.14.45 $$x \left (x-y(x)^3\right ) y'(x)=y(x) \left (y(x)^3+3 x\right )$$

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

[[_homogeneous, class G], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.591707 (sec), leaf count = 356

$\left \{\left \{y(x)\to \frac {e^{\frac {8 c_1}{3}}}{\sqrt [3]{-27 x^7+3 \sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}}+\frac {\sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}}{3^{2/3} x^2}\right \},\left \{y(x)\to \frac {\frac {i \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}\right ){}^{2/3}}{x^2}-\left (\sqrt {3}+3 i\right ) e^{\frac {8 c_1}{3}}}{2\ 3^{5/6} \sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}}\right \},\left \{y(x)\to \frac {\frac {\left (-1-i \sqrt {3}\right ) \left (-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}\right ){}^{2/3}}{x^2}+i \sqrt [3]{3} \left (\sqrt {3}+i\right ) e^{\frac {8 c_1}{3}}}{2\ 3^{2/3} \sqrt [3]{-9 x^7+\sqrt {3} \sqrt {-x^6 \left (-27 x^8+e^{8 c_1}\right )}}}\right \}\right \}$

Maple
cpu = 0.44 (sec), leaf count = 348

$\left [y \left (x \right ) = \left (\frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 \textit {\_C1}}}\right )^{\frac {1}{3}}}{3 x^{4}}+\frac {{\mathrm e}^{\frac {8 \textit {\_C1}}{3}}}{x^{4} \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 \textit {\_C1}}}\right )^{\frac {1}{3}}}\right ) x^{3}, y \left (x \right ) = \left (-\frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 \textit {\_C1}}}\right )^{\frac {1}{3}}}{6 x^{4}}-\frac {{\mathrm e}^{\frac {8 \textit {\_C1}}{3}}}{2 x^{4} \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 \textit {\_C1}}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 \textit {\_C1}}}\right )^{\frac {1}{3}}}{3 x^{4}}-\frac {{\mathrm e}^{\frac {8 \textit {\_C1}}{3}}}{x^{4} \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 \textit {\_C1}}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x^{3}, y \left (x \right ) = \left (-\frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 \textit {\_C1}}}\right )^{\frac {1}{3}}}{6 x^{4}}-\frac {{\mathrm e}^{\frac {8 \textit {\_C1}}{3}}}{2 x^{4} \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 \textit {\_C1}}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 \textit {\_C1}}}\right )^{\frac {1}{3}}}{3 x^{4}}-\frac {{\mathrm e}^{\frac {8 \textit {\_C1}}{3}}}{x^{4} \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 \textit {\_C1}}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x^{3}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-9*x^7 + Sqrt[3]*Sqrt[-(x^6*(E^(8*C[1]) - 27*x^8))])^(1/3)/(3^(2/3)*x
^2) + E^((8*C[1])/3)/(-27*x^7 + 3*Sqrt[3]*Sqrt[-(x^6*(E^(8*C[1]) - 27*x^8))])^(1
/3)}, {y[x] -> (-((3*I + Sqrt[3])*E^((8*C[1])/3)) + (I*3^(1/6)*(I + Sqrt[3])*(-9
*x^7 + Sqrt[3]*Sqrt[-(x^6*(E^(8*C[1]) - 27*x^8))])^(2/3))/x^2)/(2*3^(5/6)*(-9*x^
7 + Sqrt[3]*Sqrt[-(x^6*(E^(8*C[1]) - 27*x^8))])^(1/3))}, {y[x] -> (I*3^(1/3)*(I
+ Sqrt[3])*E^((8*C[1])/3) + ((-1 - I*Sqrt[3])*(-9*x^7 + Sqrt[3]*Sqrt[-(x^6*(E^(8
*C[1]) - 27*x^8))])^(2/3))/x^2)/(2*3^(2/3)*(-9*x^7 + Sqrt[3]*Sqrt[-(x^6*(E^(8*C[
1]) - 27*x^8))])^(1/3))}}

Maple raw input

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

Maple raw output

[y(x) = (1/3/x^4*(-27*x^4+3*(81*x^8-3*exp(8*_C1))^(1/2))^(1/3)+exp(8/3*_C1)/x^4/
(-27*x^4+3*(81*x^8-3*exp(8*_C1))^(1/2))^(1/3))*x^3, y(x) = (-1/6/x^4*(-27*x^4+3*
(81*x^8-3*exp(8*_C1))^(1/2))^(1/3)-1/2*exp(8/3*_C1)/x^4/(-27*x^4+3*(81*x^8-3*exp
(8*_C1))^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/3/x^4*(-27*x^4+3*(81*x^8-3*exp(8*_C1))^(1
/2))^(1/3)-exp(8/3*_C1)/x^4/(-27*x^4+3*(81*x^8-3*exp(8*_C1))^(1/2))^(1/3)))*x^3,
 y(x) = (-1/6/x^4*(-27*x^4+3*(81*x^8-3*exp(8*_C1))^(1/2))^(1/3)-1/2*exp(8/3*_C1)
/x^4/(-27*x^4+3*(81*x^8-3*exp(8*_C1))^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/3/x^4*(-27*x
^4+3*(81*x^8-3*exp(8*_C1))^(1/2))^(1/3)-exp(8/3*_C1)/x^4/(-27*x^4+3*(81*x^8-3*ex
p(8*_C1))^(1/2))^(1/3)))*x^3]