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 = 3.35184 (sec), leaf count = 106

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

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

\[ \left \{ {\it \_C1}+ \left (x+y \relax (x ) \right ) ^{-1}-x+{\frac {1}{ \left (x+y \relax (x ) \right ) y \relax (x ) }}=0 \right \} \] 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 + 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