Integration of a real powered rational expression Peace be upon you,
I've encountered this pretty integral
\begin{align*}
\int_0^1&\frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{x-1}dx,\\ \\
&\alpha,\beta\in\Re^+
\end{align*}
It seems much simpler than a question; but as I implied some of the known integration techniques like: Substitution, Polynomial division, Term by term integration, it was not solved.
Even I used Matlab Mupad and set the necessary assumptions and then performed the integration, but no results occurred.
Can anyone light up any idea?
 A: You may write
$$\frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{1-x} = (x^{\alpha+\beta-1}-x^{\alpha-1})\sum_{k=0}^\infty x^k,$$
then
$$\begin{align}
\int_0^1 \frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{1-x}\mathrm{d}x
&= \sum_{k=0}^\infty \int_0^1 (x^{\alpha+\beta-1}-x^{\alpha-1})x^k\mathrm{d}x\\\\
&= \sum_{k=0}^\infty \left(\frac{1}{k+\alpha+\beta} - \frac{1}{k+\alpha}\right)\\\\
&= - \psi(\alpha+\beta)+\psi(\alpha)
\end{align}$$
by the well-known series representation of the $\psi$ function DLMF: here 5.7.6.
A: Using the same approach as Olivier Oloa in his answer : 


*

*for the antiderivative $$\int \frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{x-1}\,dx=x^{\alpha } \left(\frac{1}{\alpha }-\frac{x^{\beta }}{\alpha +\beta
   }\right)-B_x(\alpha +\beta +1,0)+B_x(\alpha +1,0)$$ 

*for the integral $$\int_0^1 \frac{x^{\alpha+\beta-1}-x^{\alpha-1}}{x-1}\,dx=\psi ^{(0)}(\alpha +\beta )-\psi ^{(0)}(\alpha )$$
A: Hint: The n-th harmonic number is $~H_n=\displaystyle\int_0^1\frac{1-x^n}{1-x~~}dx.~$ Some particular values can be found here.
