4.13.13 $(y(x{)}^2+2y(x)+x)y^{\prime }(x)+y(x{)}^2(y(x)+x{)}^2+y(x)(y(x)+1)=0$

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

[_rational]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 3.35184 (sec), leaf count = 106

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

Maple ✓
cpu = 0.183 (sec), leaf count = 25

$$\{\mathit{\_}\mathit{C}\mathit{1}+{(x+y\left(x\right))}^{-1}-x+\frac{1}{(x+y\left(x\right))y\left(x\right)}=0\}$$ 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 + x^2 - x*C[1] + Sqrt[4*(x - C[1]) + (1 - x^2 + x*C[1])^2])/(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),'implicit')

Maple raw output

_C1+1/(x+y(x))-x+1/y(x)/(x+y(x)) = 0