##### 4.14.48 $$x \left (x^3+3 x^2 y(x)+y(x)^3\right ) y'(x)=y(x)^2 \left (3 x^2+y(x)^2\right )$$

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.50428 (sec), leaf count = 34

$\text {Solve}\left [\frac {y(x)^3}{3 x^3}+\frac {3 y(x)}{x}+\log \left (\frac {y(x)}{x}\right )+\log (x)=c_1,y(x)\right ]$

Maple
cpu = 0.225 (sec), leaf count = 27

$[y \left (x \right ) = {\mathrm e}^{\RootOf \left ({\mathrm e}^{3 \textit {\_Z}}+9 \,{\mathrm e}^{\textit {\_Z}}+3 \textit {\_C1} +3 \textit {\_Z} +3 \ln \left (x \right )\right )} x]$ Mathematica raw input

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

Mathematica raw output

Solve[Log[x] + Log[y[x]/x] + (3*y[x])/x + y[x]^3/(3*x^3) == C[1], y[x]]

Maple raw input

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

Maple raw output

[y(x) = exp(RootOf(exp(_Z)^3+9*exp(_Z)+3*_C1+3*_Z+3*ln(x)))*x]