Given four polynomials in $x$, called $p(x), q(x), r(x), s(x)$, such that $ \text{poly}(x) = \sum_{k=0}^N {c_k x^k},\qquad (c_k \in \mathbb{N}) \land (|c_k| \le N^2)$
Is it possible to get an easy (which will be defined later) solution to $\int {\frac {p(x) (q(x)^{1/2})} {r(x) (s(x)^{1/2})} dx}$
We define an easy solution as a solution done in polynomial time with respect to $N$ (in other words, the time it takes to get a solution is a polynomial function of $N$) and either:
(1) We get a solution of the definite integral that has arbitrary limits of integration and is accurate to $N$ decimal places. The limits may be reals with an absolute value that doesn't exceed $N^2$.
(2) We get an exact solution to the indefinite integral.
My question is, can we find some way to get an easy solution, or prove that an easy solution is impossible/possible?
MOTIVATION
This question arises from this answer to this question.
I'm attempting to determine if this integral is a potential candidate for a one-way function. It could possibly be used as a new security measure, but that may be years and years away from actual use, if it ever happens.