4.14.3 $$x \left (x^2+2 y(x)^2\right ) y'(x)=y(x) \left (2 x^2+3 y(x)^2\right )$$

ODE
$x \left (x^2+2 y(x)^2\right ) y'(x)=y(x) \left (2 x^2+3 y(x)^2\right )$ ODE Classiﬁcation

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.396208 (sec), leaf count = 163

$\left \{\left \{y(x)\to -\frac {\sqrt {-x^2-\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {-x^2-\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {-x^2+\sqrt {x^4+4 e^{2 c_1} x^6}}}{\sqrt {2}}\right \},\left \{y(x)\to \sqrt {-\frac {x^2}{2}+\frac {1}{2} \sqrt {x^4+4 e^{2 c_1} x^6}}\right \}\right \}$

Maple
cpu = 0.267 (sec), leaf count = 89

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

DSolve[x*(x^2 + 2*y[x]^2)*y'[x] == y[x]*(2*x^2 + 3*y[x]^2),y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*(x^2+2*y(x)^2)*diff(y(x),x) = (2*x^2+3*y(x)^2)*y(x), y(x))

Maple raw output

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