Let $a(x)$ and $b(x)$ be integer irreducible polynomials where $b$ is U-shaped in the interval mentioned below and has 2 distinct real zeros. The zeros of $b$ cannot be expressed by radicals. Also $b$ is the derivative of $a$ and $b$ has degree prime of type 7 mod 8.
Is it possible that there exists a nonzero integer $m$ such that the interval $[-m,m]$ contains the two zeros of $b$ and the integral from $-m$ to $m$ of the absolute value of $b(x)$ $dx$ can be given in rootform?