##### 4.11.39 $$x \left (2 x^3+y(x)\right ) y'(x)=6 y(x)^2$$

ODE
$x \left (2 x^3+y(x)\right ) y'(x)=6 y(x)^2$ ODE Classiﬁcation

[[_homogeneous, class G], _rational, [_Abel, 2nd type, class B]]

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 0.241845 (sec), leaf count = 73

$\left \{\left \{y(x)\to 2 x^3 \left (-1+\frac {2}{1-\frac {4 x^{3/2}}{\sqrt {16 x^3+c_1}}}\right )\right \},\left \{y(x)\to 2 x^3 \left (-1+\frac {2}{1+\frac {4 x^{3/2}}{\sqrt {16 x^3+c_1}}}\right )\right \}\right \}$

Maple
cpu = 0.239 (sec), leaf count = 227

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

DSolve[x*(2*x^3 + y[x])*y'[x] == 6*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> 2*x^3*(-1 + 2/(1 - (4*x^(3/2))/Sqrt[16*x^3 + C[1]]))}, {y[x] -> 2*x^3*
(-1 + 2/(1 + (4*x^(3/2))/Sqrt[16*x^3 + C[1]]))}}

Maple raw input

dsolve(x*(2*x^3+y(x))*diff(y(x),x) = 6*y(x)^2, y(x))

Maple raw output

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