##### 4.11.35 $$x (y(x)+2 x) y'(x)=x^2+x y(x)-y(x)^2$$

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.309555 (sec), leaf count = 431

$\left \{\left \{y(x)\to \text {Root}\left [32 \text {\#1}^5-80 \text {\#1}^4 x+80 \text {\#1}^3 x^2+\text {\#1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {\#1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {\#1}^5-80 \text {\#1}^4 x+80 \text {\#1}^3 x^2+\text {\#1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {\#1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {\#1}^5-80 \text {\#1}^4 x+80 \text {\#1}^3 x^2+\text {\#1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {\#1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {\#1}^5-80 \text {\#1}^4 x+80 \text {\#1}^3 x^2+\text {\#1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {\#1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {\#1}^5-80 \text {\#1}^4 x+80 \text {\#1}^3 x^2+\text {\#1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {\#1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\& ,5\right ]\right \}\right \}$

Maple
cpu = 0.291 (sec), leaf count = 63

$\left [y \left (x \right ) = -\frac {x \left (-\RootOf \left (3 \textit {\_Z}^{15}+\textit {\_Z}^{9}-2 \textit {\_C1} \,x^{3}\right )^{9}-\textit {\_C1} \,x^{3}\right )}{-\RootOf \left (3 \textit {\_Z}^{15}+\textit {\_Z}^{9}-2 \textit {\_C1} \,x^{3}\right )^{9}+2 \textit {\_C1} \,x^{3}}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> Root[E^(6*C[1])/x - x^5 + ((2*E^(6*C[1]))/x^2 + 10*x^4)*#1 + (E^(6*C[1
])/x^3 - 40*x^3)*#1^2 + 80*x^2*#1^3 - 80*x*#1^4 + 32*#1^5 & , 1]}, {y[x] -> Root
[E^(6*C[1])/x - x^5 + ((2*E^(6*C[1]))/x^2 + 10*x^4)*#1 + (E^(6*C[1])/x^3 - 40*x^
3)*#1^2 + 80*x^2*#1^3 - 80*x*#1^4 + 32*#1^5 & , 2]}, {y[x] -> Root[E^(6*C[1])/x
- x^5 + ((2*E^(6*C[1]))/x^2 + 10*x^4)*#1 + (E^(6*C[1])/x^3 - 40*x^3)*#1^2 + 80*x
^2*#1^3 - 80*x*#1^4 + 32*#1^5 & , 3]}, {y[x] -> Root[E^(6*C[1])/x - x^5 + ((2*E^
(6*C[1]))/x^2 + 10*x^4)*#1 + (E^(6*C[1])/x^3 - 40*x^3)*#1^2 + 80*x^2*#1^3 - 80*x
*#1^4 + 32*#1^5 & , 4]}, {y[x] -> Root[E^(6*C[1])/x - x^5 + ((2*E^(6*C[1]))/x^2
+ 10*x^4)*#1 + (E^(6*C[1])/x^3 - 40*x^3)*#1^2 + 80*x^2*#1^3 - 80*x*#1^4 + 32*#1^
5 & , 5]}}

Maple raw input

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

Maple raw output

[y(x) = -x*(-RootOf(3*_Z^15+_Z^9-2*_C1*x^3)^9-_C1*x^3)/(-RootOf(3*_Z^15+_Z^9-2*_
C1*x^3)^9+2*_C1*x^3)]