Elliptic/Hyperelliptic Integrals

I'm trying to find a book or website or whatever that can tell me exactly when an elliptic or hyperellptic integral (i.e. an integral of the form

\int \frac{p}{\sqrt{d}}dx

with p,d polynomials, d squarefree and of degree at least 3) is expressible in terms of elementary functions.  Everything I find is quick to point out that it can't be done in general, but doesn't tell me when it *can*, except for a few special cases which are useless to me.  Even Maple and Mathematica are no help, since they'll give answers in terms of elliptic functions even when an integral is solvable. For instance, I know that

\int \frac{y+1/2}{\sqrt{2 + 2 y + 3 y^2 + 2 y^3 + y^4}}dy = \frac{1}{4} \ln \left(\frac{f'\sqrt{d}}{4p} + f\right)

where p = y+1/2, d = 2 + 2 y + 3 y^2 + 2 y^3 + y^4, f = 3 + 4 x + 6 x^2 + 4 x^3 + 2 x^4, but both Maple and Mathematica give some horrible formula with elliptic functions.

Even if only knew some cases in which it is doable, that would be nice too.

HALP GUYZ! D:

>>4
This isn't entirely correct. If Axiom's implementation says a function does not have a simple antiderivative then it does not; but many functions which do not have simple antiderivatives will cause an error, as will some functions which do have simple antiderivatives.