Hey again,
I hope that I was able to solve my problem. This is my solution with the example of Part 1 of the question. Is that correct or did I forget anything?
$ \int \sqrt{(\sum\limits_{k=0}^{n}c_k x^k)^2} dx $
$z_1,z_2,...,z_q$ are the real roots of $\sum\limits_{k=0}^{n}c_k x^k$
sgn(x) is the signum function
$= \frac{x \sqrt{(\sum\limits_{k=0}^{n}c_k x^k)^2} (\sum\limits_{k=0}^{n}c_k x^k (\prod\limits_{m=0, m\neq k}^{n}m+1))}{(\prod\limits_{k=0}^{n}k+1) (\sum\limits_{k=0}^{n}c_k x^k)} + (\sum\limits_{p=1}^{q}sgn(x-z_p) \lim_{x \to z_p^-}(\frac{x \sqrt{(\sum\limits_{k=0}^{n}c_k x^k)^2} (\sum\limits_{k=0}^{n}c_k x^k (\prod\limits_{m=0, m\neq k}^{n}m+1))}{(\prod\limits_{k=0}^{n}k+1) (\sum\limits_{k=0}^{n}c_k x^k)}))$
Example: $n=2, c_0=-1, c_1=0, c_2=1$
$z_1 = -1, z_2 = 1$
$\int \sqrt{(\sum\limits_{k=0}^{2}c_k x^k)^2} = \sqrt{(x^2-1)^2}dx$
$=\frac{\sqrt{(x^2-1)^2} (x^3-3x)}{3*(x^2-1)} + \frac{2}{3} (sgn(x+1)+sgn(x-1))$