4.15.8 $$x (1-x y(x)) \left (1-x^2 y(x)^2\right ) y'(x)+y(x) (x y(x)+1) \left (x^2 y(x)^2+1\right )=0$$

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

[[_homogeneous, class G], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.37319 (sec), leaf count = 34

$\left \{\left \{y(x)\to -\frac {1}{x}\right \},\text {Solve}\left [\frac {1}{x y(x)}+2 \log (y(x))+c_1=x y(x),y(x)\right ]\right \}$

Maple
cpu = 0.294 (sec), leaf count = 42

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

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

Mathematica raw output

{{y[x] -> -x^(-1)}, Solve[C[1] + 2*Log[y[x]] + 1/(x*y[x]) == x*y[x], y[x]]}

Maple raw input

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

Maple raw output

[y(x) = -1/x, y(x) = exp(RootOf(-2*exp(_Z)*ln(x)-exp(2*_Z)+2*exp(_Z)*_C1+2*exp(_
Z)*_Z+1))/x]