##### 4.13.5 $$\left (x^2+y(x)^2\right ) y'(x)+2 x (y(x)+2 x)=0$$

ODE
$\left (x^2+y(x)^2\right ) y'(x)+2 x (y(x)+2 x)=0$ ODE Classiﬁcation

[[_homogeneous, class A], _exact, _rational, _dAlembert]

Book solution method
Exact equation

Mathematica
cpu = 0.521668 (sec), leaf count = 372

$\left \{\left \{y(x)\to \frac {\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}\right \},\left \{y(x)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}\right \}\right \}$

Maple
cpu = 0.078 (sec), leaf count = 417

$\left [y \left (x \right ) = \frac {\frac {\left (4-16 x^{3} \textit {\_C1}^{\frac {3}{2}}+4 \sqrt {20 \textit {\_C1}^{3} x^{6}-8 x^{3} \textit {\_C1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}-\frac {2 \textit {\_C1} \,x^{2}}{\left (4-16 x^{3} \textit {\_C1}^{\frac {3}{2}}+4 \sqrt {20 \textit {\_C1}^{3} x^{6}-8 x^{3} \textit {\_C1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}}{\sqrt {\textit {\_C1}}}, y \left (x \right ) = \frac {-\frac {\left (4-16 x^{3} \textit {\_C1}^{\frac {3}{2}}+4 \sqrt {20 \textit {\_C1}^{3} x^{6}-8 x^{3} \textit {\_C1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4}+\frac {\textit {\_C1} \,x^{2}}{\left (4-16 x^{3} \textit {\_C1}^{\frac {3}{2}}+4 \sqrt {20 \textit {\_C1}^{3} x^{6}-8 x^{3} \textit {\_C1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4-16 x^{3} \textit {\_C1}^{\frac {3}{2}}+4 \sqrt {20 \textit {\_C1}^{3} x^{6}-8 x^{3} \textit {\_C1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 \textit {\_C1} \,x^{2}}{\left (4-16 x^{3} \textit {\_C1}^{\frac {3}{2}}+4 \sqrt {20 \textit {\_C1}^{3} x^{6}-8 x^{3} \textit {\_C1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {\textit {\_C1}}}, y \left (x \right ) = \frac {-\frac {\left (4-16 x^{3} \textit {\_C1}^{\frac {3}{2}}+4 \sqrt {20 \textit {\_C1}^{3} x^{6}-8 x^{3} \textit {\_C1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4}+\frac {\textit {\_C1} \,x^{2}}{\left (4-16 x^{3} \textit {\_C1}^{\frac {3}{2}}+4 \sqrt {20 \textit {\_C1}^{3} x^{6}-8 x^{3} \textit {\_C1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4-16 x^{3} \textit {\_C1}^{\frac {3}{2}}+4 \sqrt {20 \textit {\_C1}^{3} x^{6}-8 x^{3} \textit {\_C1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 \textit {\_C1} \,x^{2}}{\left (4-16 x^{3} \textit {\_C1}^{\frac {3}{2}}+4 \sqrt {20 \textit {\_C1}^{3} x^{6}-8 x^{3} \textit {\_C1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {\textit {\_C1}}}\right ]$ Mathematica raw input

DSolve[2*x*(2*x + y[x]) + (x^2 + y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((2^(1/3)*x^2)/(E^(3*C[1]) - 4*x^3 + Sqrt[E^(6*C[1]) - 8*E^(3*C[1])*x
^3 + 20*x^6])^(1/3)) + (E^(3*C[1]) - 4*x^3 + Sqrt[E^(6*C[1]) - 8*E^(3*C[1])*x^3
+ 20*x^6])^(1/3)/2^(1/3)}, {y[x] -> (2^(1/3)*(2 + (2*I)*Sqrt[3])*x^2 + I*2^(2/3)
*(I + Sqrt[3])*(E^(3*C[1]) - 4*x^3 + Sqrt[E^(6*C[1]) - 8*E^(3*C[1])*x^3 + 20*x^6
])^(2/3))/(4*(E^(3*C[1]) - 4*x^3 + Sqrt[E^(6*C[1]) - 8*E^(3*C[1])*x^3 + 20*x^6])
^(1/3))}, {y[x] -> ((1 - I*Sqrt[3])*x^2)/(2^(2/3)*(E^(3*C[1]) - 4*x^3 + Sqrt[E^(
6*C[1]) - 8*E^(3*C[1])*x^3 + 20*x^6])^(1/3)) - ((1 + I*Sqrt[3])*(E^(3*C[1]) - 4*
x^3 + Sqrt[E^(6*C[1]) - 8*E^(3*C[1])*x^3 + 20*x^6])^(1/3))/(2*2^(1/3))}}

Maple raw input

dsolve((x^2+y(x)^2)*diff(y(x),x)+2*x*(2*x+y(x)) = 0, y(x))

Maple raw output

[y(x) = (1/2*(4-16*x^3*_C1^(3/2)+4*(20*_C1^3*x^6-8*x^3*_C1^(3/2)+1)^(1/2))^(1/3)
-2*_C1*x^2/(4-16*x^3*_C1^(3/2)+4*(20*_C1^3*x^6-8*x^3*_C1^(3/2)+1)^(1/2))^(1/3))/
_C1^(1/2), y(x) = (-1/4*(4-16*x^3*_C1^(3/2)+4*(20*_C1^3*x^6-8*x^3*_C1^(3/2)+1)^(
1/2))^(1/3)+_C1*x^2/(4-16*x^3*_C1^(3/2)+4*(20*_C1^3*x^6-8*x^3*_C1^(3/2)+1)^(1/2)
)^(1/3)-1/2*I*3^(1/2)*(1/2*(4-16*x^3*_C1^(3/2)+4*(20*_C1^3*x^6-8*x^3*_C1^(3/2)+1
)^(1/2))^(1/3)+2*_C1*x^2/(4-16*x^3*_C1^(3/2)+4*(20*_C1^3*x^6-8*x^3*_C1^(3/2)+1)^
(1/2))^(1/3)))/_C1^(1/2), y(x) = (-1/4*(4-16*x^3*_C1^(3/2)+4*(20*_C1^3*x^6-8*x^3
*_C1^(3/2)+1)^(1/2))^(1/3)+_C1*x^2/(4-16*x^3*_C1^(3/2)+4*(20*_C1^3*x^6-8*x^3*_C1
^(3/2)+1)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(4-16*x^3*_C1^(3/2)+4*(20*_C1^3*x^6-8*
x^3*_C1^(3/2)+1)^(1/2))^(1/3)+2*_C1*x^2/(4-16*x^3*_C1^(3/2)+4*(20*_C1^3*x^6-8*x^
3*_C1^(3/2)+1)^(1/2))^(1/3)))/_C1^(1/2)]