Let $s(x) = \frac{\sin(\pi x)}{\pi x}$ be the normalized sinc function, $a_j' = \frac{a_j}{\pi}$ and $c = \frac{b+\sum_j a_j}{\pi}$. Then you want to compute $\pi\int_{-\infty}^\infty c s(cx)\prod_j s(a_j' x)dx$. This is equal to the value of the Fourier transform of the integrand at zero. The Fourier transform of the integrand is
\[
r\left(\frac{f}{c}\right) \ast \left[\pi\cdot\frac{1}{a_1'}r\left(\frac{f}{a_1'}\right) \ast \cdots \ast \frac{1}{a_n'}r\left(\frac{f}{a_n'}\right)\right],
\]
where $r$ denotes the indicator function of the interval $[-1,1]$.

Note that the bracketed term has compact support contained in the interval $\left[-\sum_j a_j', \sum_j a_j'\right]$, so in particular contained in $[-c,c]$ regardless of $b$. So the value at zero of this outermost convolution is just the integral of the bracketed term. We can compute this integral by Fourier transforming again and evaluating the result at zero. But this is just $\pi\cdot\prod_j s(a_j' \cdot 0) = \pi$.

productof several char fcns of symmetrical intervals, which is just the FT of the char fcn of the smallest among them. If this is in the right direction for you, I could elaborate. $\endgroup$ – paul garrett Mar 1 '17 at 23:11