##### 4.15.16 $$2 x \left (x^3+y(x)^4\right ) y'(x)=y(x) \left (x^3+2 y(x)^4\right )$$

ODE
$2 x \left (x^3+y(x)^4\right ) y'(x)=y(x) \left (x^3+2 y(x)^4\right )$ ODE Classiﬁcation

[[_homogeneous, class G], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.310873 (sec), leaf count = 161

$\left \{\left \{y(x)\to -\frac {\sqrt {c_1 x^2-x^{3/2} \sqrt {4+c_1{}^2 x}}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {c_1 x^2-x^{3/2} \sqrt {4+c_1{}^2 x}}}{\sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {x^{3/2} \sqrt {4+c_1{}^2 x}+c_1 x^2}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {x^{3/2} \sqrt {4+c_1{}^2 x}+c_1 x^2}}{\sqrt {2}}\right \}\right \}$

Maple
cpu = 0.15 (sec), leaf count = 293

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

DSolve[2*x*(x^3 + y[x]^4)*y'[x] == y[x]*(x^3 + 2*y[x]^4),y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(2*x*(x^3+y(x)^4)*diff(y(x),x) = (x^3+2*y(x)^4)*y(x), y(x))

Maple raw output

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