##### 4.13.43 $$x \left (x^2+y(x)^2\right ) y'(x)=y(x) \left (x^4+x^2+y(x)^2\right )$$

ODE
$x \left (x^2+y(x)^2\right ) y'(x)=y(x) \left (x^4+x^2+y(x)^2\right )$ ODE Classiﬁcation

[[_homogeneous, class D], _rational]

Book solution method
Homogeneous equation, $$xy'(x)=x f(x) g(u)+y(x)$$

Mathematica
cpu = 0.394823 (sec), leaf count = 44

$\left \{\left \{y(x)\to -x \sqrt {W\left (e^{x^2+2 c_1}\right )}\right \},\left \{y(x)\to x \sqrt {W\left (e^{x^2+2 c_1}\right )}\right \}\right \}$

Maple
cpu = 0.104 (sec), leaf count = 26

$\left [y \left (x \right ) = {\mathrm e}^{-\frac {\LambertW \left ({\mathrm e}^{x^{2}} {\mathrm e}^{2 \textit {\_C1}}\right )}{2}+\frac {x^{2}}{2}+\textit {\_C1}} x\right ]$ Mathematica raw input

DSolve[x*(x^2 + y[x]^2)*y'[x] == y[x]*(x^2 + x^4 + y[x]^2),y[x],x]

Mathematica raw output

{{y[x] -> -(x*Sqrt[ProductLog[E^(x^2 + 2*C[1])]])}, {y[x] -> x*Sqrt[ProductLog[E
^(x^2 + 2*C[1])]]}}

Maple raw input

dsolve(x*(x^2+y(x)^2)*diff(y(x),x) = (x^2+x^4+y(x)^2)*y(x), y(x))

Maple raw output

[y(x) = exp(-1/2*LambertW(exp(x^2)*exp(_C1)^2)+1/2*x^2+_C1)*x]