Claim on Wikipedia in connection with integrability and Risch's algorithm. Any references?

Question

The Wikipedia article on Risch's algorithm makes this interesting statement:

the following algebraic function has an elementary antiderivative: $$f(x) = \frac{x}{\sqrt{x^4 + 10 x^2 - 96 x - 71}}$$ [...] But if the constant term 71 is changed to 72, it is not possible to represent the antiderivative in terms of elementary functions.

For the first part of the statement the antiderivative is given, together with a (quite cool) reference to a post by the late Bronstein. But for the second part of the statement no reference is given (as often happens in Wikipedia; so the second part is just a claim). I tried to check the book by Geddes & al but didn't manage to find that statement there.

Does anyone know any reference to a proof that if 71 is changed to 72 no antiderivative exists that can be expressed in terms of elementary functions?

Answer 1

This integral was solved by Chebyshev here https://archive.org/details/117744684_001/page/n11/mode/2up

Ultimately because fractional expansion of this function in denominator has a specific form we know integral is elementary and we construct a subtitution from expansion coeff. See https://math.stackexchange.com/a/3933268/1013030

What is funny full proof for that was only given by Zolotarev.

Answer 2

Comment:May be this idea helps:

$x^4+10x^2-96x-71=(x^2+5)^2-96(x+1)=$

Let:

$x^2+5=u\Rightarrow 2xdx=du\Rightarrow dx\frac {du}{2x}=\frac{du}{2\sqrt{u-5}}$

Putting this in integrand you get:

$$ \frac{dx}{x^4+10x^2-96x-71}=\frac{du}{[u-96(\sqrt{u-5}+1)](2\sqrt{u-5})}$$

Which can be transformed to two fractions.This is not possible if you replace 71 by 72.