I'm looking to simplify the integral
$\int\nolimits_0^{\infty}\dfrac{(t^b+1)^n}{(1+t)^{nb+2}} dt$.
(This arises out of the sum of a bunch of Beta functions, ie $\displaystyle\sum_{i=0}^{n} \binom{n}{i} B(1+ib,1+(n-i)b)$ with $b$ a probably irrational constant, $\approx 1.64677$. I already know about this version of the simplification.)
I'm not very experienced in simplifying integrals like this, and I would welcome any help or suggestions you would provide.