$$ \displaystyle\int_a^{b} \left(\frac{1}{\frac{\sqrt{x^2 + 356 x + 322}}{10000}}\right)^{4×5} \left(\frac{\sqrt{x^2 + 356 x + 322}}{10000}\right)^{1000} \frac{\left(π^{(-1 - 4×5 + 10000)} \displaystyle\binom{10000}{1 + 4 k} \right)}{\left(\frac{\sqrt{x^2 + 356 x + 322}}{10000}\right)} dx$$
I'm having a hard time coming up with a non-complex integral for this function. I know I can use euler substitution to handle the sqrt function but apart from that I seem stuck.
If you get rid of all constant, you end with the problem of the antiderivative $$I_n=\int \Big[x^2 + 356 x + 322\Big]^{n+\frac 12}\,dx$$ Your case corresponds to $I_{489}$.
Write $x^2 + 356 x + 322=(x-\alpha)(x-\beta)$ and use Euler third substitution $$\sqrt{(x-\alpha)(x-\beta)}=(x-\alpha)t$$ which implies $$x=\frac{\alpha-\beta t^2}{1-t^2}\quad \implies \quad dx=2(\beta-\alpha)\frac{ t }{\left(1-t^2\right)^2}\,dt$$
$$I_n=2(\beta -\alpha )^{2 (n+1)}\int t^{2 (n+1)}\,\left(1-t^2\right)^{-(2 n+3)}\,dt$$ This looks very much like an incomplete beta function.