4.12.9 $(x^2+6xy(x)+3)y^{\prime }(x)+3y(x{)}^2+2xy(x)+2x=0$

ODE
$$(x^2+6xy(x)+3)y^{\prime }(x)+3y(x{)}^2+2xy(x)+2x=0$$ ODE Classification

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

Book solution method
Exact equation

Mathematica ✓
cpu = 0.011937 (sec), leaf count = 83

$$\{\{y(x)\to -\frac{\sqrt{36c_{1}x+x^4-12x^3+6x^2+9}+x^2+3}{6x}\},\{y(x)\to -\frac{-\sqrt{36c_{1}x+x^4-12x^3+6x^2+9}+x^2+3}{6x}\}\}$$

Maple ✓
cpu = 0.014 (sec), leaf count = 25

$$\{x^{2}y\left(x\right)+3x{\left(y\left(x\right)\right)}^2+x^2+\mathit{\_}\mathit{C}\mathit{1}+3y\left(x\right)=0\}$$ 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] -> -(3 + x^2 + Sqrt[9 + 6*x^2 - 12*x^3 + x^4 + 36*x*C[1]])/(6*x)}, {y[x] 
-> -(3 + x^2 - Sqrt[9 + 6*x^2 - 12*x^3 + x^4 + 36*x*C[1]])/(6*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),'implicit')

Maple raw output

x^2*y(x)+3*x*y(x)^2+x^2+_C1+3*y(x) = 0