4.12.3 $x(-2y(x)+x+1)y^{\prime }(x)+y(x)(y(x)-2x+1)=0$

ODE
$$x(-2y(x)+x+1)y^{\prime }(x)+y(x)(y(x)-2x+1)=0$$ ODE Classification

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

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 15.0879 (sec), leaf count = 457

$$\{\{y(x)\to -\frac{\sqrt[3]{2}x}{\sqrt[3]{27c_1^2{x}^2+\sqrt{(27c_1^2{x}^2+27c_1^{2}x){}^2-108c_1^3{x}^3}+27c_1^{2}x}}-\frac{\sqrt[3]{27c_1^2{x}^2+\sqrt{(27c_1^2{x}^2+27c_1^{2}x){}^2-108c_1^3{x}^3}+27c_1^{2}x}}{3\sqrt[3]{2}{c}_1}-x-1\},\{y(x)\to \frac{(1+i\sqrt{3})x}{2^{2/3}\sqrt[3]{27c_1^2{x}^2+\sqrt{(27c_1^2{x}^2+27c_1^{2}x){}^2-108c_1^3{x}^3}+27c_1^{2}x}}+\frac{(1-i\sqrt{3})\sqrt[3]{27c_1^2{x}^2+\sqrt{(27c_1^2{x}^2+27c_1^{2}x){}^2-108c_1^3{x}^3}+27c_1^{2}x}}{6\sqrt[3]{2}{c}_1}-x-1\},\{y(x)\to \frac{(1-i\sqrt{3})x}{2^{2/3}\sqrt[3]{27c_1^2{x}^2+\sqrt{(27c_1^2{x}^2+27c_1^{2}x){}^2-108c_1^3{x}^3}+27c_1^{2}x}}+\frac{(1+i\sqrt{3})\sqrt[3]{27c_1^2{x}^2+\sqrt{(27c_1^2{x}^2+27c_1^{2}x){}^2-108c_1^3{x}^3}+27c_1^{2}x}}{6\sqrt[3]{2}{c}_1}-x-1\}\}$$

Maple ✓
cpu = 0.054 (sec), leaf count = 74

$$\{\frac{7}{9}\ln \left(\frac{-189-189x}{4+4x-8y\left(x\right)}\right)-\frac{7}{6}\ln (-126\frac{1+x+y\left(x\right)}{1+x-2y\left(x\right)})+\frac{7}{18}\ln (-378\frac{y\left(x\right)}{5+5x-10y\left(x\right)})-\frac{7\ln (1+x)}{9}+\frac{7\ln \left(x\right)}{18}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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),'implicit')

Maple raw output

7/9*ln((-189-189*x)/(4+4*x-8*y(x)))-7/6*ln(-126*(1+x+y(x))/(1+x-2*y(x)))+7/18*ln
(-378*y(x)/(5+5*x-10*y(x)))-7/9*ln(1+x)+7/18*ln(x)-_C1 = 0