##### 4.12.9 $$\left (x^2+6 x y(x)+3\right ) y'(x)+3 y(x)^2+2 x y(x)+2 x=0$$

ODE
$\left (x^2+6 x y(x)+3\right ) y'(x)+3 y(x)^2+2 x y(x)+2 x=0$ ODE Classiﬁcation

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

Book solution method
Exact equation

Mathematica
cpu = 0.385642 (sec), leaf count = 83

$\left \{\left \{y(x)\to -\frac {x^2+\sqrt {x^4-12 x^3+6 x^2+36 c_1 x+9}+3}{6 x}\right \},\left \{y(x)\to -\frac {x^2-\sqrt {x^4-12 x^3+6 x^2+36 c_1 x+9}+3}{6 x}\right \}\right \}$

Maple
cpu = 0.018 (sec), leaf count = 71

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

DSolve[2*x + 2*x*y[x] + 3*y[x]^2 + (3 + x^2 + 6*x*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -1/6*(3 + x^2 + Sqrt[9 + 6*x^2 - 12*x^3 + x^4 + 36*x*C[1]])/x}, {y[x]
-> -1/6*(3 + x^2 - Sqrt[9 + 6*x^2 - 12*x^3 + x^4 + 36*x*C[1]])/x}}

Maple raw input

dsolve((3+6*x*y(x)+x^2)*diff(y(x),x)+2*x+2*x*y(x)+3*y(x)^2 = 0, y(x))

Maple raw output

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