Elliptic integral and pseudo-elliptic integral from Wikipedia.

studentstrug

Hi all.

I was reading this Wikipedia article: http://en.wikipedia.org/wiki/Risch_algorithm

I have a couple of questions about [tex]\int \frac{x}{\sqrt{x^4 + 10x^2 - 96x - 71}} \, dx[/tex] and [tex]\int \frac{x}{\sqrt{x^4 + 10x^2 - 96x - 72}} \, dx[/tex] discussed in the article.

How the heck did they get the elementary form of the first integral? I tried differentiating the answer given - using WolframAlpha - to check it and got a big mess with lots of radicals, not the integrand. I haven't been brave enough to try and simplify yet. Is their answer actually true?

I also have no idea how they constructed these two examples. Would it have just been by guessing different forms, putting them into a CAS and see which one had an answer in terms of elementary functions? WolframAlpha is no help. I suspect the construction of these two integrals is related to how to get the answer to the first.

There is no reference so I'm assuming they were just made up by the author .....

Anyway, if anyone can shed some light on the answers I'd be obliged.

Thanks in advance.

studentstrug

OK, the integral is correct, the derivative of it gives the correct integrand - I made a data entry error with WolframAlpha (1001 instead of 10001).

But I still don't get HOW they constructed these two examples - how thay knew that 71 would work but 72 would not work. I believe it has something to do with a particular expression involving D = x^4 + 10x^2 - 96x - 71 being a non-trivial unit in the function field Q(x, sqrt(D)) .... (which makes epsilon more sense to me than talking about the Galois group of D ....)

If someone could explain it so that it makes some sense to a humble first year university student I'd be obliged.