4.12.4 $x(-2y(x)-x+1)y^{\prime }(x)+y(x)(y(x)+2x+1)=0$

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 = 14.9599 (sec), leaf count = 451

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

Maple ✓
cpu = 0.032 (sec), leaf count = 76

$$\{\frac{7}{9}\ln \left(\frac{189-189x}{-4+4x+8y\left(x\right)}\right)-\frac{7}{6}\ln (-126\frac{x-y\left(x\right)-1}{-1+x+2y\left(x\right)})+\frac{7}{18}\ln \left(378\frac{y\left(x\right)}{-5+5x+10y\left(x\right)}\right)-\frac{7\ln (-1+x)}{9}+\frac{7\ln \left(x\right)}{18}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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 + Sqr
t[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 + 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])*(2
7*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)*(2
7*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*(x-y(x)-1)/(-1+x+2*y(x)))+7/18*l
n(378*y(x)/(-5+5*x+10*y(x)))-7/9*ln(-1+x)+7/18*ln(x)-_C1 = 0