##### 4.15.4 $$x \left (-2 x y(x)^3-x y(x)^2+2\right ) y'(x)+2 y(x)+1=0$$

ODE
$x \left (-2 x y(x)^3-x y(x)^2+2\right ) y'(x)+2 y(x)+1=0$ ODE Classiﬁcation

[_rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]]

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 1.03003 (sec), leaf count = 42

$\text {Solve}\left [\frac {1}{64} \left (-4 y(x)^2+4 y(x)-\frac {16}{2 x y(x)+x}-2 \log (8 y(x)+4)+3\right )=c_1,y(x)\right ]$

Maple
cpu = 0.317 (sec), leaf count = 42

$\left [y \left (x \right ) = \frac {{\mathrm e}^{\RootOf \left (x \,{\mathrm e}^{3 \textit {\_Z}}-4 x \,{\mathrm e}^{2 \textit {\_Z}}+8 x \textit {\_C1} \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}+3 x \,{\mathrm e}^{\textit {\_Z}}+16\right )}}{2}-\frac {1}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

Solve[(3 - 2*Log[4 + 8*y[x]] + 4*y[x] - 4*y[x]^2 - 16/(x + 2*x*y[x]))/64 == C[1]
, y[x]]

Maple raw input

dsolve(x*(2-x*y(x)^2-2*x*y(x)^3)*diff(y(x),x)+1+2*y(x) = 0, y(x))

Maple raw output

[y(x) = 1/2*exp(RootOf(x*exp(_Z)^3-4*x*exp(_Z)^2+8*x*_C1*exp(_Z)+2*_Z*x*exp(_Z)+
3*x*exp(_Z)+16))-1/2]