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In Piskunov's Calulus (P375 & P385) & Hardy's Integration of Functions of a Single Variable (P48) mention is made of Chebyshev's theorem on the integration of binomial differentials however no proof for the general case is offered. I'm wondering if anybody has a proof of this theorem in general & hopefully a few references in English that discuss this theorem. My suspicions are that this theorem can be proven using methods of differential algebra (whether it actually is in the literature or not is another story) though I'm really hoping for something along the lines Chebyshev himself would have followed (along the strands thread out in Hardy's book), thanks for your time...

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  1. All works of Chebyshev are available in French (and Russian, of course).

  2. There is a recent book MR2106657 M. Bronstein, Symbolic integration. I., which must contain this (sorry I could not check; do not have it beside me).

Two other books are: MR0617377 J. Davenport, On the integration of algebraic functions, and J. F. Ritt, Integration in Finite Terms. Liouville's Theory of Elementary Methods.

Third. The idea is simple. In all cases, except those 3 when the integral is known, the Riemann surface of the integrand is completely ramified over 3 points. Such surfaces are of positive genus. Of course this is not a proof but just a hint, one needs to study the monodromy group.

However it is really interesting how Chebyshev proved it: they say Chebyshev did not use even complex numbers, not speaking of the monodromy group:-)

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