I tried it on Maple and Mathematica, but neither is giving me an answer to this integral
$$\int_0^{\infty } \frac{\sqrt{\left(u^2+1\right) \left(u^6+4 u^4-4 u^3-4 u+4\right)}}{\left(u^3+1\right)^2} \, du$$
Interesting in http://mathworld.wolfram.com/FoliumofDescartes.html, WolframAlpha gives an answer
I am guessing this is non-elementary and must use numerical methods?
EDIT: What method is used to calculate this integral? Is this elliptic?
There is a secret society of time-traveling humans (coming from 5000s) working on that website. They can even integrate $e^{x^2-\sin(\cos(x))}$ in closed form!
Jokes aside, the fact that they gave the result as $4.917488...a$ (note the dots) gives me the impression that it is numerical integration.