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:

Look at the FullSimplify command; specifically the examples with ComplexityFunction. Then just tell Mathematica that Elliptic functions should be given a horrible score during the simplification.

>>2
Meh, still didn't work.  Thanks though, I didn't know that you could do that with ComplexityFunction.

OP here.  If anyone's interested, I sort of found the answer.  Risch's algorithm gives a definitive answer to the question of whether any particular antiderivative is doable, but it's not fully implemented in any computer algebra system.  Wiki says Axiom has it implemented well enough to definitively say that an integral isn't doable, but it doesn't say when it *is*.  Unfortunately, Axiom is proving to be a PITA to install on windows.

>>4
IIRC, when I installed it some years ago it was just download and unzip... PITA to google for information on it on, though. Stupid generic word names.

>>5

On unix it's no problem, but windows is awful.  You have to download a bunch of huge programs to essentially run Ubuntu inside windows that end up taking 2 gigs of HD space and then run axiom inside that.  Then (for some reason) when you actually get into axiom your keyboard is all scrambled ¯\(°_o)/¯. Luckily, it's easy to add into Sage and run from there. (use install_package("fricas-1.0.3.p0"))

Their website, with details on how to install it in windows: http://www.axiom-developer.org/index.html

>>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.

>>7
Right.  That's what I was trying to say.

>>6
Huh. I checked, and I do have a native windows console version 0.1.4 from 2005. 50MB install file, 170 MB installed.
Looks clean enough, can't imagine how they got from there to where they are now.