4.11.26 $x(y(x)+4)y^{\prime }(x)=y(x{)}^2+2y(x)+2x$

ODE
$$x(y(x)+4)y^{\prime }(x)=y(x{)}^2+2y(x)+2x$$ ODE Classification

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

Book solution method
Homogeneous equation, special

Mathematica ✓
cpu = 0.0220779 (sec), leaf count = 96

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

Maple ✓
cpu = 0.033 (sec), leaf count = 74

$$\{{(4+y\left(x\right))}^{-1}-1\sqrt{x}{(4+x)}^{-\frac{3}{2}}\frac{1}{\sqrt{\mathit{\_}\mathit{C}\mathit{1}-4{(4+x)}^{-1}}}-{(4+x)}^{-1}=0,{(4+y\left(x\right))}^{-1}+1\sqrt{x}{(4+x)}^{-\frac{3}{2}}\frac{1}{\sqrt{\mathit{\_}\mathit{C}\mathit{1}-4{(4+x)}^{-1}}}-{(4+x)}^{-1}=0\}$$ 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) - (x/(4 + x))^(3/2)/(x*Sqrt[(-4 + (4 + x)*C[1])/(4 
+ x)]))^(-1)}, {y[x] -> -4 + ((4 + x)^(-1) + (x/(4 + x))^(3/2)/(x*Sqrt[(-4 + (4 
+ x)*C[1])/(4 + x)]))^(-1)}}

Maple raw input

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

Maple raw output

1/(4+y(x))+x^(1/2)/(4+x)^(3/2)/(_C1-4/(4+x))^(1/2)-1/(4+x) = 0, 1/(4+y(x))-x^(1/
2)/(4+x)^(3/2)/(_C1-4/(4+x))^(1/2)-1/(4+x) = 0