##### 4.12.3 $$x (-2 y(x)+x+1) y'(x)+y(x) (y(x)-2 x+1)=0$$

ODE
$x (-2 y(x)+x+1) y'(x)+y(x) (y(x)-2 x+1)=0$ ODE Classiﬁcation

[_rational, [_Abel, 2nd type, class B]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 10.2234 (sec), leaf count = 457

$\left \{\left \{y(x)\to -\frac {\sqrt [3]{2} x}{\sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}-\frac {\sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}{3 \sqrt [3]{2} c_1}-x-1\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}-x-1\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 c_1{}^2 x^2+\sqrt {\left (27 c_1{}^2 x^2+27 c_1{}^2 x\right ){}^2-108 c_1{}^3 x^3}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}-x-1\right \}\right \}$

Maple
cpu = 0.109 (sec), leaf count = 499

$\left [y \left (x \right ) = \frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 x^{2} \textit {\_C1} +160 x \textit {\_C1} +80 \textit {\_C1} -x}{\textit {\_C1}}}-20 x -20\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}{40 \textit {\_C1}}+\frac {3 x 5^{\frac {2}{3}}}{40 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 x^{2} \textit {\_C1} +160 x \textit {\_C1} +80 \textit {\_C1} -x}{\textit {\_C1}}}-20 x -20\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}-x -1, y \left (x \right ) = -\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 x^{2} \textit {\_C1} +160 x \textit {\_C1} +80 \textit {\_C1} -x}{\textit {\_C1}}}-20 x -20\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}{80 \textit {\_C1}}-\frac {3 x 5^{\frac {2}{3}}}{80 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 x^{2} \textit {\_C1} +160 x \textit {\_C1} +80 \textit {\_C1} -x}{\textit {\_C1}}}-20 x -20\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}-x -1-\frac {i \sqrt {3}\, \left (\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 x^{2} \textit {\_C1} +160 x \textit {\_C1} +80 \textit {\_C1} -x}{\textit {\_C1}}}-20 x -20\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}{40 \textit {\_C1}}-\frac {3 x 5^{\frac {2}{3}}}{40 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 x^{2} \textit {\_C1} +160 x \textit {\_C1} +80 \textit {\_C1} -x}{\textit {\_C1}}}-20 x -20\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 x^{2} \textit {\_C1} +160 x \textit {\_C1} +80 \textit {\_C1} -x}{\textit {\_C1}}}-20 x -20\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}{80 \textit {\_C1}}-\frac {3 x 5^{\frac {2}{3}}}{80 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 x^{2} \textit {\_C1} +160 x \textit {\_C1} +80 \textit {\_C1} -x}{\textit {\_C1}}}-20 x -20\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}-x -1+\frac {i \sqrt {3}\, \left (\frac {3 \,5^{\frac {1}{3}} \left (x \left (\sqrt {5}\, \sqrt {\frac {80 x^{2} \textit {\_C1} +160 x \textit {\_C1} +80 \textit {\_C1} -x}{\textit {\_C1}}}-20 x -20\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}{40 \textit {\_C1}}-\frac {3 x 5^{\frac {2}{3}}}{40 \left (x \left (\sqrt {5}\, \sqrt {\frac {80 x^{2} \textit {\_C1} +160 x \textit {\_C1} +80 \textit {\_C1} -x}{\textit {\_C1}}}-20 x -20\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -1 - x - (2^(1/3)*x)/(27*x*C[1]^2 + 27*x^2*C[1]^2 + Sqrt[-108*x^3*C[1]
^3 + (27*x*C[1]^2 + 27*x^2*C[1]^2)^2])^(1/3) - (27*x*C[1]^2 + 27*x^2*C[1]^2 + Sq
rt[-108*x^3*C[1]^3 + (27*x*C[1]^2 + 27*x^2*C[1]^2)^2])^(1/3)/(3*2^(1/3)*C[1])},
{y[x] -> -1 - x + ((1 + I*Sqrt[3])*x)/(2^(2/3)*(27*x*C[1]^2 + 27*x^2*C[1]^2 + Sq
rt[-108*x^3*C[1]^3 + (27*x*C[1]^2 + 27*x^2*C[1]^2)^2])^(1/3)) + ((1 - I*Sqrt[3])
*(27*x*C[1]^2 + 27*x^2*C[1]^2 + Sqrt[-108*x^3*C[1]^3 + (27*x*C[1]^2 + 27*x^2*C[1
]^2)^2])^(1/3))/(6*2^(1/3)*C[1])}, {y[x] -> -1 - x + ((1 - I*Sqrt[3])*x)/(2^(2/3
)*(27*x*C[1]^2 + 27*x^2*C[1]^2 + Sqrt[-108*x^3*C[1]^3 + (27*x*C[1]^2 + 27*x^2*C[
1]^2)^2])^(1/3)) + ((1 + I*Sqrt[3])*(27*x*C[1]^2 + 27*x^2*C[1]^2 + Sqrt[-108*x^3
*C[1]^3 + (27*x*C[1]^2 + 27*x^2*C[1]^2)^2])^(1/3))/(6*2^(1/3)*C[1])}}

Maple raw input

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

Maple raw output

[y(x) = 3/40/_C1*5^(1/3)*(x*(5^(1/2)*((80*_C1*x^2+160*_C1*x+80*_C1-x)/_C1)^(1/2)
-20*x-20)*_C1^2)^(1/3)+3/40*x*5^(2/3)/(x*(5^(1/2)*((80*_C1*x^2+160*_C1*x+80*_C1-
x)/_C1)^(1/2)-20*x-20)*_C1^2)^(1/3)-x-1, y(x) = -3/80/_C1*5^(1/3)*(x*(5^(1/2)*((
80*_C1*x^2+160*_C1*x+80*_C1-x)/_C1)^(1/2)-20*x-20)*_C1^2)^(1/3)-3/80*x*5^(2/3)/(
x*(5^(1/2)*((80*_C1*x^2+160*_C1*x+80*_C1-x)/_C1)^(1/2)-20*x-20)*_C1^2)^(1/3)-x-1
-1/2*I*3^(1/2)*(3/40/_C1*5^(1/3)*(x*(5^(1/2)*((80*_C1*x^2+160*_C1*x+80*_C1-x)/_C
1)^(1/2)-20*x-20)*_C1^2)^(1/3)-3/40*x*5^(2/3)/(x*(5^(1/2)*((80*_C1*x^2+160*_C1*x
+80*_C1-x)/_C1)^(1/2)-20*x-20)*_C1^2)^(1/3)), y(x) = -3/80/_C1*5^(1/3)*(x*(5^(1/
2)*((80*_C1*x^2+160*_C1*x+80*_C1-x)/_C1)^(1/2)-20*x-20)*_C1^2)^(1/3)-3/80*x*5^(2
/3)/(x*(5^(1/2)*((80*_C1*x^2+160*_C1*x+80*_C1-x)/_C1)^(1/2)-20*x-20)*_C1^2)^(1/3
)-x-1+1/2*I*3^(1/2)*(3/40/_C1*5^(1/3)*(x*(5^(1/2)*((80*_C1*x^2+160*_C1*x+80*_C1-
x)/_C1)^(1/2)-20*x-20)*_C1^2)^(1/3)-3/40*x*5^(2/3)/(x*(5^(1/2)*((80*_C1*x^2+160*
_C1*x+80*_C1-x)/_C1)^(1/2)-20*x-20)*_C1^2)^(1/3))]