I'd like to symbolically integrate the expression $\int_0^1{\|r'\left(t\right)\|_2\,dt}$ where $r$ is a function $\mathbb{R} \rightarrow \mathbb{R}^2$ (so the expression is the arc length of the curve $r$). It'd be nice if this would work for any function that can be integrated and differentiated, but I'd already be pretty happy if it worked for polynomials.
Expanding the definition of the Euclidean norm yields
$$\int_0^1{\sqrt{\left(r'_1\left(t\right)\right)^2+\left(r'_2\left(t\right)\right)^2}\,dt}$$
where $r'_1$ and $r'_2$ denote the component functions of $r'$. I've done some research and it seems that there is no general solution for the integral of a square root of an arbitrary polynomial, but maybe the special form of this one makes it possible? Unfortunately, it doesn't look like any of the popular integration identities applies here.
If it turns out to be impossible, I'll be able to use numerical integration, but since I want to use this expression in a smooth nonlinear optimization problem, it'd be a lot easier if there was a symbolic solution to this integration, so I'd be very grateful for any pointers on how to do this, if it is possible.