4.13.44 $$x \left (-x^2+y(x)^2+1\right ) y'(x)+y(x) \left (x^2-y(x)^2+1\right )=0$$

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

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

Book solution method
Homogeneous equation, special

Mathematica
cpu = 0.444724 (sec), leaf count = 83

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

Maple
cpu = 0.061 (sec), leaf count = 114

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[1/(1/y(x)^2-1/(x^2-1)) = -x*(x+1)^(1/2)*(x-1)^(1/2)/(_C1+2/(x-1)-2/(x+1))^(1/2)
-1/2*(x-1)*(x+1), 1/(1/y(x)^2-1/(x^2-1)) = x*(x+1)^(1/2)*(x-1)^(1/2)/(_C1+2/(x-1
)-2/(x+1))^(1/2)-1/2*(x-1)*(x+1)]