ODE
\[ 3 x y'(x)+2 (y(x)+1)^{3/2}-3 y(x)=0 \] ODE Classification
[_separable]
Book solution method
Clairaut’s equation and related types, main form
Mathematica ✓
cpu = 0.0369379 (sec), leaf count = 49
\[\text {Solve}\left [3 c_1=\text {RootSum}\left [2 \text {$\#$1}^3-3 \text {$\#$1}^2+3\& ,\frac {\log \left (\sqrt {y(x)+1}-\text {$\#$1}\right )}{\text {$\#$1}-1}\& \right ]+\log (x),y(x)\right ]\]
Maple ✓
cpu = 0.035 (sec), leaf count = 31
\[ \left \{ \ln \left ( x \right ) +\int ^{y \left ( x \right ) }\!3\, \left ( \left ( 2\,{\it \_a}+2 \right ) \sqrt {{\it \_a}+1}-3\,{\it \_a} \right ) ^{-1}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[-3*y[x] + 2*(1 + y[x])^(3/2) + 3*x*y'[x] == 0,y[x],x]
Mathematica raw output
Solve[3*C[1] == Log[x] + RootSum[3 - 3*#1^2 + 2*#1^3 & , Log[-#1 + Sqrt[1 + y[x]]]/(-1 + #1) & ], y[x]]
Maple raw input
dsolve(2*(1+y(x))^(3/2)+3*x*diff(y(x),x)-3*y(x) = 0, y(x),'implicit')
Maple raw output
ln(x)+Intat(3/((2*_a+2)*(_a+1)^(1/2)-3*_a),_a = y(x))+_C1 = 0