Take the product $$(x-a_1)^{b_1} \cdots (x-a_n)^{b_n}$$ of $n$ factors, with real roots $a_1 \ldots a_n$ and real exponents $b_1 \ldots b_n$.
Its antiderivative for $n=2$ and $n=3$ can be expressed in terms of the hypergeometric functions. Are there special functions that express the integral $$\int \! (x-a_1)^{b_1} \cdots (x-a_n)^{b_n} \, dx$$ for higher $n$, either for specific values or generally?
Edit:
Perhaps it is a start to consider more promising special cases.
Let $\alpha = (a_1, \ldots, a_n)$, $\beta = (b_1, \ldots, b_n)$, $\kappa = (k_1, \ldots, k_n)$ be multi-indices and take the particular form of the integral $$ \int \! x^m \prod_{i=1}^{n} (x - a_i)^{b_i} \, dx = \sum_{\kappa \ge 0} \binom{\beta}{\kappa} \, (-1)^{|\kappa|} \, \alpha^\kappa \int \! x^{m + |\beta| - |\kappa|} \, dx \;. $$ The power series form on the right-hand side is how the question arose, see my comment about a related question. Now fix $m$ s.t. $m+|\beta|+1=0$, then $$ \int \! x^{-|\beta|-1} \prod_{i=1}^{n} (x - a_i)^{b_i} \, dx = \log(x) - \sum_{\substack{\kappa \ge 0\\|\kappa|>0}} \binom{\beta}{\kappa} \, \frac{(-1)^{|\kappa|} \, \alpha^\kappa}{|\kappa|} \, x^{-|\kappa|} \;. $$ This seems to me a multi-index version of a hypergeometric function. Does anyone recognise it?