##### 4.9.3 $$\sqrt {x^3+1} y'(x)=\sqrt {y(x)^3+1}$$

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]