##### 4.12.10 $$x^3+3 x (2 y(x)+x) y'(x)+3 y(x) (y(x)+2 x)=0$$

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

[_exact, _rational, [_Abel, 2nd type, class B]]

Book solution method
Exact equation

Mathematica
cpu = 0.319339 (sec), leaf count = 75

$\left \{\left \{y(x)\to -\frac {3 x^2+\sqrt {-3 x^5+9 x^4+36 c_1 x}}{6 x}\right \},\left \{y(x)\to \frac {-3 x^2+\sqrt {-3 x^5+9 x^4+36 c_1 x}}{6 x}\right \}\right \}$

Maple
cpu = 0.016 (sec), leaf count = 63

$\left [y \left (x \right ) = \frac {-3 x^{2}+\sqrt {-3 x^{5}+9 x^{4}-12 x \textit {\_C1}}}{6 x}, y \left (x \right ) = -\frac {3 x^{2}+\sqrt {-3 x^{5}+9 x^{4}-12 x \textit {\_C1}}}{6 x}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -1/6*(3*x^2 + Sqrt[9*x^4 - 3*x^5 + 36*x*C[1]])/x}, {y[x] -> (-3*x^2 +
Sqrt[9*x^4 - 3*x^5 + 36*x*C[1]])/(6*x)}}

Maple raw input

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

Maple raw output

[y(x) = 1/6*(-3*x^2+(-3*x^5+9*x^4-12*_C1*x)^(1/2))/x, y(x) = -1/6*(3*x^2+(-3*x^5
+9*x^4-12*_C1*x)^(1/2))/x]