Prove or give reference for $\int \frac{1}{\sum _{k=1}^n x^{k-1} \binom{n-1}{k-1}} \, dx=\frac{(x+1)^{2-n}}{2-n}$

Question

Mathematica knows that: $$\int \frac{1}{\sum _{k=1}^n x^{k-1} \binom{n-1}{k-1}} \, dx=\frac{(x+1)^{2-n}}{2-n}$$

I have never seen this in a Mathematical handbook before. To me this explains the essence of the integration of fractions with polynomials in the denominator.

Could you point me to a book or a formula collection, or even give a proof if you find it worth the effort?