##### 4.11.26 $$x (y(x)+4) y'(x)=y(x)^2+2 y(x)+2 x$$

ODE
$x (y(x)+4) y'(x)=y(x)^2+2 y(x)+2 x$ ODE Classiﬁcation

[_rational, [_Abel, 2nd type, class B]]

Book solution method
Homogeneous equation, special

Mathematica
cpu = 0.286372 (sec), leaf count = 84

$\left \{\left \{y(x)\to -4+\frac {1}{\frac {1}{x+4}-\frac {\sqrt {x}}{(x+4)^{3/2} \sqrt {-\frac {4}{x+4}+c_1}}}\right \},\left \{y(x)\to -4+\frac {1}{\frac {1}{x+4}+\frac {\sqrt {x}}{(x+4)^{3/2} \sqrt {-\frac {4}{x+4}+c_1}}}\right \}\right \}$

Maple
cpu = 0.059 (sec), leaf count = 145

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

DSolve[x*(4 + y[x])*y'[x] == 2*x + 2*y[x] + y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -4 + ((4 + x)^(-1) - Sqrt[x]/((4 + x)^(3/2)*Sqrt[-4/(4 + x) + C[1]]))^
(-1)}, {y[x] -> -4 + ((4 + x)^(-1) + Sqrt[x]/((4 + x)^(3/2)*Sqrt[-4/(4 + x) + C[
1]]))^(-1)}}

Maple raw input

dsolve(x*(4+y(x))*diff(y(x),x) = 2*x+2*y(x)+y(x)^2, y(x))

Maple raw output

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