Evaluate \(\int_{0}^{1} \frac{x^{n}}{\sqrt{x^{3}+1}} dx\)

How to evaluate
\begin{equation}
\int\limits_{0}^{1} \frac{x^{n}}{\sqrt{x^{3}+1}} dx
\end{equation}
was a question posed at Mathematics Stack Exchange. Here is my solution.

Let \(y=x^{3}\)
\begin{align}
\int\limits_{0}^{1} \frac{x^{n}}{\sqrt{x^{3}+1}} dx
&= \frac{1}{3} \int\limits_{0}^{1} \frac{y^{(n-2)/3}}{\sqrt{y+1}} dy \\
&= \frac{1}{3} \frac{\Gamma(\frac{n+1}{3})\Gamma(1)}{\Gamma(\frac{n+4}{3})}\,{}_{2}\mathrm{F}_{1}\left(\frac{1}{2},\frac{n+1}{3};\frac{n+4}{3};-1\right) \\
&= \frac{1}{n+1} \,{}_{2}\mathrm{F}_{1}\left(\frac{1}{2},\frac{n+1}{3};\frac{n+4}{3};-1\right)
\end{align}

We used the analytic continuation of Gauss’s hypergeometric function
\begin{equation}
{}_{2}\mathrm{F}_{1}(a,b;c;z)
= \frac{\Gamma(c)}{\Gamma(b)\Gamma(c – b)} \int\limits_{0}^{1} t^{b-1} (1-t)^{c-b-1} (1-zt)^{-a} dt
\end{equation}
For \(\mathrm{Re}\, c \gt \mathrm{Re}\, b \gt 0 \, , \, |\mathrm{arg}(1-z)| \lt \pi\)

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