##### 4.15.19 $$x \left (x^3+y(x)^5\right ) y'(x)=y(x) \left (x^3-y(x)^5\right )$$

ODE
$x \left (x^3+y(x)^5\right ) y'(x)=y(x) \left (x^3-y(x)^5\right )$ ODE Classiﬁcation

[[_homogeneous, class G], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.311688 (sec), leaf count = 141

$\left \{\left \{y(x)\to \text {Root}\left [4 \text {\#1}^5 x-4 \text {\#1}^4 c_1-x^4\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {\#1}^5 x-4 \text {\#1}^4 c_1-x^4\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {\#1}^5 x-4 \text {\#1}^4 c_1-x^4\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {\#1}^5 x-4 \text {\#1}^4 c_1-x^4\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [4 \text {\#1}^5 x-4 \text {\#1}^4 c_1-x^4\& ,5\right ]\right \}\right \}$

Maple
cpu = 0.409 (sec), leaf count = 36

$\left [\ln \left (x \right )-\textit {\_C1} -\frac {5 \ln \left (\frac {y \left (x \right )}{x^{\frac {3}{5}}}\right )}{2}+\frac {5 \ln \left (-\frac {-4 y \left (x \right )^{5}+x^{3}}{x^{3}}\right )}{8} = 0\right ]$ Mathematica raw input

DSolve[x*(x^3 + y[x]^5)*y'[x] == y[x]*(x^3 - y[x]^5),y[x],x]

Mathematica raw output

{{y[x] -> Root[-x^4 - 4*C[1]*#1^4 + 4*x*#1^5 & , 1]}, {y[x] -> Root[-x^4 - 4*C[1
]*#1^4 + 4*x*#1^5 & , 2]}, {y[x] -> Root[-x^4 - 4*C[1]*#1^4 + 4*x*#1^5 & , 3]},
{y[x] -> Root[-x^4 - 4*C[1]*#1^4 + 4*x*#1^5 & , 4]}, {y[x] -> Root[-x^4 - 4*C[1]
*#1^4 + 4*x*#1^5 & , 5]}}

Maple raw input

dsolve(x*(x^3+y(x)^5)*diff(y(x),x) = (x^3-y(x)^5)*y(x), y(x))

Maple raw output

[ln(x)-_C1-5/2*ln(y(x)/x^(3/5))+5/8*ln(-(-4*y(x)^5+x^3)/x^3) = 0]