Let $\alpha>1$, $n \in \mathbb{N}$ and $q\geq0$. Which methods is possible to use to solve this integral?
$$\int_{1}^{\alpha} x^n(x^2-1)^{q-\frac{1}{2}}dx$$
I tried using the computer for especific values of $n$, but wolfram only calculate for $n=5$ giving the result
$$\frac{((α^2 - 1)^{q+\frac{1}{2}} (α^2 (2 q + 1) (α^2 (2 q + 3) + 4) + 8))}{((2 q + 1) (2 q + 3) (2 q + 5))}.$$
I think this integral is connected with some special function, but I don't know which one.
Thank you for the patience and the reading.
Let $t=\frac{1}{x^2},~~x=\frac{1}{\sqrt{t}},~~~dx=-\frac{1}{2}t^{-\frac{3}2}$, and plug in the substitution.
$$I=\frac{1}{2}\int_\frac{1}{\sqrt{\alpha}}^1 t^{-1-q-\frac{n}{2}}(1-t)^{q-\frac{1}{2}}dt=\frac{1}{2}\cdot\left(\text{B}(-q-\frac{n}{2},q+\frac{1}{2})-\text{B}(-q-\frac{n}{2},q+\frac{1}{2},\frac{1}{\sqrt{\alpha}})\right)$$
where $\text{B}(-q-\frac{n}{2},q+\frac{1}{2},\frac{1}{\sqrt{\alpha}})$ is incomplete beta function.