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 = 0.0106324 (sec), leaf count = 73
\[\left \{\left \{y(x)\to 2 x^3 \left (\frac {2}{1-\frac {4 x^{3/2}}{\sqrt {c_1+16 x^3}}}-1\right )\right \},\left \{y(x)\to 2 x^3 \left (\frac {2}{\frac {4 x^{3/2}}{\sqrt {c_1+16 x^3}}+1}-1\right )\right \}\right \}\]
Maple ✓
cpu = 0.02 (sec), leaf count = 33
\[ \left \{ \ln \left ( x \right ) -{\it \_C1}-{\frac {2}{3}\ln \left ( {\frac {-2\,{x}^{3}+y \left ( x \right ) }{{x}^{3}}} \right ) }+{\frac {1}{3}\ln \left ( {\frac {y \left ( x \right ) }{{x}^{3}}} \right ) }=0 \right \} \] 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),'implicit')
Maple raw output
ln(x)-_C1-2/3*ln((-2*x^3+y(x))/x^3)+1/3*ln(y(x)/x^3) = 0