What's the fastest way to get an exact value for a product of (powers of polynomials)?

Question

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.

Answer 1

I'd forget about the integral for a moment. And deal with the problem as computing the polynomial $$ R(x) = \displaystyle \prod_k{\left(p_k(x)\right)^{c_k}} $$ which you can do using evaluation-then-interpolation (e.g. my other answer). Afterwords, a definite integration of $R(x)$ should be straightforward operation.

Another approach to compute $R(x)$ is to consider the following Jordan block matrix: $$ A = \begin{pmatrix} C(p_1(x)^{c_1}) && 0 && \ldots && 0 \\ 0 && C(p_2(x)^{c_2}) && 0 && \ldots && 0 \\ 0 && \ldots && \ldots && \ldots && \\ 0 && \ldots && 0 && \ldots && C(p_n(x)^{c_n}) \end{pmatrix}$$ where $C(p_k(x)^{c_k})$ is the block companion matrix of $p_k(x)$. Then $R(x) = \operatorname{charpoly}(A)$ which can be computed using fast linear algebra methods.