ODE
\[ \left (2 x^2+3 y(x)^2\right ) y'(x)+x (y(x)+3 x)=0 \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.401697 (sec), leaf count = 66
\[\text {Solve}\left [c_1=\text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}+1\& ,\frac {3 \text {$\#$1}^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )+2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )}{3 \text {$\#$1}^2+1}\& \right ]+3 \log (x),y(x)\right ]\]
Maple ✓
cpu = 0.031 (sec), leaf count = 35
\[\left [y \left (x \right ) = \RootOf \left (\int _{}^{\textit {\_Z}}\frac {3 \textit {\_a}^{2}+2}{\textit {\_a}^{3}+\textit {\_a} +1}d \textit {\_a} +3 \ln \left (x \right )+3 \textit {\_C1} \right ) x\right ]\] Mathematica raw input
DSolve[x*(3*x + y[x]) + (2*x^2 + 3*y[x]^2)*y'[x] == 0,y[x],x]
Mathematica raw output
Solve[C[1] == 3*Log[x] + RootSum[1 + #1 + #1^3 & , (2*Log[-#1 + y[x]/x] + 3*Log[-#1 + y[x]/x]*#1^2)/(1 + 3*#1^2) & ], y[x]]
Maple raw input
dsolve((2*x^2+3*y(x)^2)*diff(y(x),x)+x*(3*x+y(x)) = 0, y(x))
Maple raw output
[y(x) = RootOf(Intat(1/(_a^3+_a+1)*(3*_a^2+2),_a = _Z)+3*ln(x)+3*_C1)*x]