Suppose we have two integers $a$ and $b$. Also, suppose we have polynomials in $x$, $p_k(x)$. Finally, suppose we have a sequence of integers, where an integer in the sequence is denoted by $c_k$.
What's the fastest way to get an exact value for $\int_a^b{\left(\displaystyle \prod_k{\left(p_k(x)\right)^{c_k}}\right) dx}$, with the $c_k$s large?
This is a more complicated version of this question. Perhaps "What's the fastest way to get an exact value for integrate a power of a polynomial?" may help with ideas.