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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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Interesting question. I remember the theorem, and the proof of the easy direction (by explicit substitutions), but never saw the proof of the hard direction. I'm assuming it's the latter that you are asking about. –  user53153 Feb 24 '13 at 0:29
    
Yeah the substitution direction is offered in Piskunov. Also on mathoverflow –  sponsoredwalk Feb 24 '13 at 8:56
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