ODE
\[ \sqrt {x^3+1} y'(x)=\sqrt {y(x)^3+1} \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.28313 (sec), leaf count = 46
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\text {$\#$1} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-\text {$\#$1}^3\right )\& \right ]\left [x \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-x^3\right )+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.014 (sec), leaf count = 28
\[\left [\int \frac {1}{\sqrt {x^{3}+1}}d x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a}^{3}+1}}d \textit {\_a} \right )+\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[Sqrt[1 + x^3]*y'[x] == Sqrt[1 + y[x]^3],y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Hypergeometric2F1[1/3, 1/2, 4/3, -#1^3]*#1 & ][C[1] + x*Hypergeometric2F1[1/3, 1/2, 4/3, -x^3]]}}
Maple raw input
dsolve(diff(y(x),x)*(x^3+1)^(1/2) = (1+y(x)^3)^(1/2), y(x))
Maple raw output
[Int(1/(x^3+1)^(1/2),x)-Intat(1/(_a^3+1)^(1/2),_a = y(x))+_C1 = 0]