##### 4.12.8 $$x (3 y(x)+2 x) y'(x)+3 (y(x)+x)^2=0$$

ODE
$x (3 y(x)+2 x) y'(x)+3 (y(x)+x)^2=0$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, [_Abel, 2nd type, class B]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.325485 (sec), leaf count = 71

$\left \{\left \{y(x)\to -\frac {4 x^2+\sqrt {-2 x^4+6 e^{4 c_1}}}{6 x}\right \},\left \{y(x)\to \frac {-4 x^2+\sqrt {-2 x^4+6 e^{4 c_1}}}{6 x}\right \}\right \}$

Maple
cpu = 0.064 (sec), leaf count = 63

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

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

Mathematica raw output

{{y[x] -> -1/6*(4*x^2 + Sqrt[6*E^(4*C[1]) - 2*x^4])/x}, {y[x] -> (-4*x^2 + Sqrt[
6*E^(4*C[1]) - 2*x^4])/(6*x)}}

Maple raw input

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

Maple raw output

[y(x) = (-2/3*x^2*_C1-1/6*(-2*_C1^2*x^4+6)^(1/2))/_C1/x, y(x) = (-2/3*x^2*_C1+1/
6*(-2*_C1^2*x^4+6)^(1/2))/_C1/x]