Comment #4 on issue 1393 by fredrik.johansson: integrate(x**2 *
sqrt(5-x**2), x) can't do it yet
http://code.google.com/p/sympy/issues/detail?id=1393
Quadratics are by far most important, and the only ones that can be done
(generally)
using elementary functions.
For polynomials of degree 3 or 4, elliptic integrals are needed
(http://en.wikipedia.org/wiki/Elliptic_integral)
If you have the spare time, symbolic elliptic integrals would be a fine
addition to
SymPy, but I suggest focusing on the elementary case first.
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