TITLE: $\int 5\sqrt{x^3+2x}\, \textrm{d}x$
QUESTION [7 upvotes]: I can not find any method to solve this, why do the typical methods fail? 
I have tried subtituion, parts, strange subitution, etc. 
How does one solve this integral?

REPLY [3 votes]: This is an elliptic integral. The elliptic integrals represent solutions for the arc length of an elliptic arc and they cannot be expressed in terms of elementary functions.
I do not know if the above statement is proved, or closed forms in terms of elementary functions exist but not yet discovered. Anyways, among the people who tried to find a solution was Euler and Legendre.
The elliptic integrals can be expressed (reduced) to Legendre canonical forms (there are three of them).