4.12.4 \(x (-2 y(x)-x+1) y'(x)+y(x) (y(x)+2 x+1)=0\)

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 = 14.9599 (sec), leaf count = 451

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

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

\[ \left \{ {\frac {7}{9}\ln \left ({\frac {189-189\,x}{-4+4\,x+8\,y \relax (x ) }} \right ) }-{\frac {7}{6}\ln \left (-126\,{\frac {x-y \relax (x ) -1}{-1+x+2\,y \relax (x ) }} \right ) }+{\frac {7}{18}\ln \left (378\,{\frac {y \relax (x ) }{-5+5\,x+10\,y \relax (x ) }} \right ) }-{\frac {7\,\ln \left (-1+x \right ) }{9}}+{\frac {7\,\ln \relax (x ) }{18}}-{\it \_C1}=0 \right \} \] 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 + 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