4.14.39 $$y(x) \left (5 x^2+2 y(x)^2\right ) y'(x)+x \left (x^2+5 y(x)^2\right )=0$$

ODE
$y(x) \left (5 x^2+2 y(x)^2\right ) y'(x)+x \left (x^2+5 y(x)^2\right )=0$ ODE Classiﬁcation

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

Book solution method
Exact equation

Mathematica
cpu = 0.483512 (sec), leaf count = 159

$\left \{\left \{y(x)\to -\frac {\sqrt {-5 x^2-\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {-5 x^2-\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {-5 x^2+\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {-5 x^2+\sqrt {23 x^4+2 e^{4 c_1}}}}{\sqrt {2}}\right \}\right \}$

Maple
cpu = 0.25 (sec), leaf count = 125

$\left [y \left (x \right ) = -\frac {\sqrt {-10 x^{2} \textit {\_C1} -2 \sqrt {23 x^{4} \textit {\_C1}^{2}+2}}}{2 \sqrt {\textit {\_C1}}}, y \left (x \right ) = \frac {\sqrt {-10 x^{2} \textit {\_C1} -2 \sqrt {23 x^{4} \textit {\_C1}^{2}+2}}}{2 \sqrt {\textit {\_C1}}}, y \left (x \right ) = -\frac {\sqrt {-10 x^{2} \textit {\_C1} +2 \sqrt {23 x^{4} \textit {\_C1}^{2}+2}}}{2 \sqrt {\textit {\_C1}}}, y \left (x \right ) = \frac {\sqrt {-10 x^{2} \textit {\_C1} +2 \sqrt {23 x^{4} \textit {\_C1}^{2}+2}}}{2 \sqrt {\textit {\_C1}}}\right ]$ Mathematica raw input

DSolve[x*(x^2 + 5*y[x]^2) + y[x]*(5*x^2 + 2*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[-5*x^2 - Sqrt[2*E^(4*C[1]) + 23*x^4]]/Sqrt[2])}, {y[x] -> Sqrt[
-5*x^2 - Sqrt[2*E^(4*C[1]) + 23*x^4]]/Sqrt[2]}, {y[x] -> -(Sqrt[-5*x^2 + Sqrt[2*
E^(4*C[1]) + 23*x^4]]/Sqrt[2])}, {y[x] -> Sqrt[-5*x^2 + Sqrt[2*E^(4*C[1]) + 23*x
^4]]/Sqrt[2]}}

Maple raw input

dsolve((5*x^2+2*y(x)^2)*y(x)*diff(y(x),x)+x*(x^2+5*y(x)^2) = 0, y(x))

Maple raw output

[y(x) = -1/2*(-10*x^2*_C1-2*(23*_C1^2*x^4+2)^(1/2))^(1/2)/_C1^(1/2), y(x) = 1/2*
(-10*x^2*_C1-2*(23*_C1^2*x^4+2)^(1/2))^(1/2)/_C1^(1/2), y(x) = -1/2*(-10*x^2*_C1
+2*(23*_C1^2*x^4+2)^(1/2))^(1/2)/_C1^(1/2), y(x) = 1/2*(-10*x^2*_C1+2*(23*_C1^2*
x^4+2)^(1/2))^(1/2)/_C1^(1/2)]