4.7.4 $$\left (x^2+1\right ) y'(x)=y(x)^2-2 x \left (y(x)^2+1\right ) y(x)+1$$

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

[_rational, _Abel]

Book solution method
Abel ODE, Second kind

Mathematica
cpu = 0.810284 (sec), leaf count = 161

$\text {Solve}\left [c_1=\frac {i \left (x \left (\sqrt [4]{\frac {\left (x^2+1\right ) \left (y(x)^2+1\right )}{(x y(x)-1)^2}} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};-\frac {(x+y(x))^2}{(x y(x)-1)^2}\right )-2\right )+y(x) \left (\sqrt [4]{\frac {\left (x^2+1\right ) \left (y(x)^2+1\right )}{(x y(x)-1)^2}} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};-\frac {(x+y(x))^2}{(x y(x)-1)^2}\right )+2 x^2\right )\right )}{2 (x y(x)-1) \sqrt [4]{-\frac {\left (x^2+1\right ) \left (y(x)^2+1\right )}{(x y(x)-1)^2}}},y(x)\right ]$

Maple
cpu = 0.078 (sec), leaf count = 85

$\left [\textit {\_C1} +\frac {x}{\left (\left (\frac {1}{x}+\frac {x^{2}}{\frac {x^{4} y \left (x \right )}{x^{2}+1}-\frac {x^{3}}{x^{2}+1}}\right )^{2}+1\right )^{\frac {1}{4}}}+\frac {\left (x +y \left (x \right )\right ) \hypergeom \left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x +y \left (x \right )\right )^{2}}{\left (x y \left (x \right )-1\right )^{2}}\right )}{2 x y \left (x \right )-2} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

Solve[C[1] == ((I/2)*(x*(-2 + Hypergeometric2F1[1/2, 5/4, 3/2, -((x + y[x])^2/(-
1 + x*y[x])^2)]*(((1 + x^2)*(1 + y[x]^2))/(-1 + x*y[x])^2)^(1/4)) + y[x]*(2*x^2
+ Hypergeometric2F1[1/2, 5/4, 3/2, -((x + y[x])^2/(-1 + x*y[x])^2)]*(((1 + x^2)*
(1 + y[x]^2))/(-1 + x*y[x])^2)^(1/4))))/((-1 + x*y[x])*(-(((1 + x^2)*(1 + y[x]^2
))/(-1 + x*y[x])^2))^(1/4)), y[x]]

Maple raw input

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

Maple raw output

[_C1+x/((1/x+x^2/(x^4/(x^2+1)*y(x)-1/(x^2+1)*x^3))^2+1)^(1/4)+(x+y(x))*hypergeom
([1/2, 5/4],[3/2],-(x+y(x))^2/(x*y(x)-1)^2)/(2*x*y(x)-2) = 0]