On the integrals $\int_{-1}^0 \sqrt[2n+1]{x-\sqrt[2n+1]x} \mathrm dx$

For $n \in \mathbb{N}$ we have \begin{align} I (n) &= \int \limits_{-1}^0 \left[x - x^{\frac{1}{2n+1}}\right]^{\frac{1}{2n+1}} \mathrm{d} x \stackrel{x = -y}{=} \int \limits_0^1 \left[y^{\frac{1}{2n+1}} - y\right]^{\frac{1}{2n+1}} \mathrm{d} y = \int \limits_0^1 y^{\frac{1}{(2n+1)^2}}\left[1 - y^{\frac{2n}{2n+1}}\right]^{\frac{1}{2n+1}} \mathrm{d} y \\ &\hspace{-10pt}\stackrel{y = t^{\frac{2n+1}{2n}}}{=} \frac{2n+1}{2n} \int \limits_0^1 t^{\frac{n+1}{n (2n+1)}} (1-t)^{\frac{1}{2n+1}} \mathrm{d}t = \frac{2n+1}{2n} \operatorname{B}\left(\frac{n+1}{n(2n+1)}+1,\frac{1}{2n+1} + 1\right) \, . \end{align} Using $\Gamma(x+1) = x \Gamma(x)$ we can rewrite this result to find $$ I (n) = \frac{\operatorname{B} \left(\frac{1}{n} - \frac{1}{2n+1}, \frac{1}{2n+1}\right)}{2 (2n+1)} = \frac{1}{2(2n+1)} \frac{\operatorname{\Gamma}\left(\frac{1}{n} - \frac{1}{2n+1}\right) \operatorname{\Gamma}\left(\frac{1}{2n+1}\right)}{\operatorname{\Gamma}\left(\frac{1}{n}\right)}$$ for $n \in \mathbb{N}$. In particular, $$ I(1) = \frac{\operatorname{\Gamma}\left(\frac{2}{3}\right) \operatorname{\Gamma}\left(\frac{1}{3}\right)}{6} = \frac{\pi}{6 \sin \left(\frac{\pi}{3}\right)} = \frac{\pi}{3 \sqrt{3}} \,.$$ Moreover, we obtain $$ \lim_{n \to \infty} I (n) \stackrel{\Gamma(x) \, \stackrel{x \to 0}{\sim} \, \frac{1}{x}}{=} \lim_{n \to \infty} \frac{1}{2(2n+1)} \frac{\frac{n(2n+1)}{n+1} (2n+1)}{n} = \lim_{n \to \infty} \frac{2n+1}{2(n+1)} = 1 \, .$$

Too long for a comment.

After ComplexYetTrivial's answer, we have $$I_n=\frac{\Gamma \left(\frac{2 (n+1)}{2 n+1}\right) \Gamma \left(\frac{n+1}{n(2 n+1)}\right)}{2 \Gamma \left(\frac{1}{n}\right)}$$ Expanding as series $$I_n=1-\frac{1}{2 n}+\frac{12-\pi ^2}{24 n^2}+O\left(\frac{1}{n^3}\right)$$ which is not "too bad" even for $n=1$; this would give $1-\frac{\pi ^2}{24}\approx 0.588766$ while $\frac{\pi}{3 \sqrt{3}}\approx 0.604600$.

For a few values of $n$ $$\left( \begin{array}{ccc} n & \text{approximation} & \text{exact} \\ 1 & 0.588766 & 0.604600 \\ 2 & 0.772192 & 0.774848 \\ 3 & 0.843196 & 0.843965 \\ 4 & 0.880548 & 0.880851 \\ 5 & 0.903551 & 0.903695 \\ 6 & 0.919132 & 0.919210 \\ 7 & 0.930383 & 0.930429 \\ 8 & 0.938887 & 0.938916 \\ 9 & 0.945540 & 0.945560 \end{array} \right)$$