Is there an upper bound for the following integral in terms of $c$ and $A$ ?
$\int_{-\infty}^{+\infty}\frac{(1+|x+c|)^A}{(1+|x|)^B} dx$,
where $c, A\in \mathbb R$ and $B\in \mathbb R^+$.
Here $A$ is fixed and $B$ is a parameter and can be chosen as large as possible. Simplest case is when $c=0$. In this case the integral is absolutely convergent (by choosing $B$ appropriately) and thus bounded by some absolute constant.
Hint:
It is possible to get an exact expression. Due to the fact that there is an arbitrary additive parameter $c$, the problem is not simpler than the evaluation of the indefinite integral. Then it is advisable to get rid of the absolute values by splitting the domain in the three intervals delimited by $0$ and $-c$.
So, with a shift of the variable it suffices to consider
$$\int\frac{(x+c)^a}{x^b}dx.$$
When $a$ is an integer, you develop the binomial and end-up with a finite sum of powers. Otherwise, I don't think you can escape the incomplete Beta integral.