4.12.30 $x(2xy(x)+1)y^{\prime }(x)+y(x)(3xy(x)+2)=0$

ODE
$$x(2xy(x)+1)y^{\prime }(x)+y(x)(3xy(x)+2)=0$$ ODE Classification

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

Book solution method
Homogeneous equation, isobaric equation

Mathematica ✓
cpu = 0.0131273 (sec), leaf count = 69

$$\{\{y(x)\to -\frac{\sqrt{x^2(4c_1+x)}+x^{3/2}}{2x^{5/2}}\},\{y(x)\to \frac{\sqrt{x^2(4c_1+x)}-x^{3/2}}{2x^{5/2}}\}\}$$

Maple ✓
cpu = 0.012 (sec), leaf count = 20

$$\{\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}+\ln \left(xy\left(x\right)(1+xy\left(x\right))\right)=0\}$$ Mathematica raw input

DSolve[y[x]*(2 + 3*x*y[x]) + x*(1 + 2*x*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*(1+2*x*y(x))*diff(y(x),x)+(2+3*x*y(x))*y(x) = 0, y(x),'implicit')

Maple raw output

ln(x)-_C1+ln(x*y(x)*(1+x*y(x))) = 0