##### 4.13.13 $$\left (y(x)^2+2 y(x)+x\right ) y'(x)+y(x)^2 (y(x)+x)^2+y(x) (y(x)+1)=0$$

ODE
$\left (y(x)^2+2 y(x)+x\right ) y'(x)+y(x)^2 (y(x)+x)^2+y(x) (y(x)+1)=0$ ODE Classiﬁcation

[_rational]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.938622 (sec), leaf count = 106

$\left \{\left \{y(x)\to -\frac {x^2+\sqrt {\left (-x^2+c_1 x+1\right ){}^2+4 (x-c_1)}-c_1 x-1}{2 (x-c_1)}\right \},\left \{y(x)\to \frac {-x^2+\sqrt {\left (-x^2+c_1 x+1\right ){}^2+4 (x-c_1)}+c_1 x+1}{2 (x-c_1)}\right \}\right \}$

Maple
cpu = 0.195 (sec), leaf count = 120

$\left [y \left (x \right ) = \frac {-x \textit {\_C1} +2 x^{2}-2+\sqrt {\textit {\_C1}^{2} x^{2}-4 \textit {\_C1} \,x^{3}+4 x^{4}+4 x \textit {\_C1} -8 x^{2}-8 \textit {\_C1} +16 x +4}}{2 \textit {\_C1} -4 x}, y \left (x \right ) = -\frac {x \textit {\_C1} -2 x^{2}+\sqrt {\textit {\_C1}^{2} x^{2}-4 \textit {\_C1} \,x^{3}+4 x^{4}+4 x \textit {\_C1} -8 x^{2}-8 \textit {\_C1} +16 x +4}+2}{2 \left (\textit {\_C1} -2 x \right )}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -1/2*(-1 + x^2 - x*C[1] + Sqrt[4*(x - C[1]) + (1 - x^2 + x*C[1])^2])/(
x - C[1])}, {y[x] -> (1 - x^2 + x*C[1] + Sqrt[4*(x - C[1]) + (1 - x^2 + x*C[1])^
2])/(2*(x - C[1]))}}

Maple raw input

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

Maple raw output

[y(x) = 1/2/(_C1-2*x)*(-x*_C1+2*x^2-2+(_C1^2*x^2-4*_C1*x^3+4*x^4+4*_C1*x-8*x^2-8
*_C1+16*x+4)^(1/2)), y(x) = -1/2*(x*_C1-2*x^2+(_C1^2*x^2-4*_C1*x^3+4*x^4+4*_C1*x
-8*x^2-8*_C1+16*x+4)^(1/2)+2)/(_C1-2*x)]