Let $0<\alpha\leq\frac{1}{2}$ a fixed real number. I wondered if it is possible to evaluate the sequence of definite integrals $$\int\_0^1\left(\sum\_{k=0}^n (f(x))^k\right)^{\alpha}dx\tag{1}$$
for some example of a continuous function and positive $f(x)>0$ for all $0<x<1$, and $f(x)\neq\text{constant}$.



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> **Question.** Do you know an example of evaluation of $(1)$, the evaluation $\forall n\geq 1$, for a choice of $0<\alpha\leq\frac{1}{2}$ and a function $f(x)$ satisfying previous requirements? **Many thanks.**
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If the example that I evoke is in the literature, please refer it and I try to search and read the example from the literature. 



Example of evaluation of $\int_0^1\left(\sum_{k=0}^n (f(x))^k\right)^{\alpha}dx$, for some choice of $f(x)$ satisfying certain requirements
If you allow for special functions, you of course allow for quite some beasts. Generally, your expression is
$$
I = \int\_{0}^{1} dx \left( \sum\_{k=0}^{n} (f(x))^k \right)^{\alpha }
= \int\_{0}^{1} dx \left( \frac{1-(f(x))^{n+1} }{1-f(x)} \right)^{\alpha }
$$
Let's choose $f(x)=x^2 $ and $\alpha =1/2$, and substitute integration variable $x=\sin t$. This yields
$$
I = \int\_{0}^{\pi /2} dt \sqrt{1-\sin^{2n+2} t} \\
=\frac{\pi }{2}\cdot {}\_{n+2}F\_{n+1}
\left( -\frac{1}{2},\frac{1}{2n+2},\frac{3}{2n+2},\ldots,\frac{2n+1}{2n+2};\frac{1}{n+1},\frac{2}{n+1},\ldots,\frac{n}{n+1},1;1 \right)
$$


