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I am reading about elliptic integrals, and there is an article showing how to transform general form $\int R(x, \sqrt{P(x)})\,\mathrm{d}x$ with $P(x)$ being a polynomial of degree 3 or 4. It boils down to the three type of elliptic integrals (ie, non-elementary):

$$ \int \frac{1}{\sqrt{P(x)}}\,\mathrm{d}x\tag{$I_0$} $$

$$ \int \frac{x^2}{\sqrt{P(x)}}\,\mathrm{d}x\tag{$I_2$} $$ and $$ \int \frac{1}{(x-b)\sqrt{P(x)}}\,\mathrm{d}x\tag{$H_1$} $$

It is explicitly mentioned that the $I_1$ integral, $\int \frac{x}{\sqrt{P(x)}}\,\mathrm{d}x$, is elementary, but didn't show how to integrate it. I searched the web but found nothing similar. Could someone give a hint how to reduce it to elementary integral?

TIA

Shine
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    Which article ? – Angina Seng Feb 02 '18 at 06:23
  • Are you sure that this is not in the case of repeated roots for $P(x)=0$ ? – Claude Leibovici Feb 02 '18 at 06:45
  • I think we're all a little skeptical about that integral being elementary (without some additional assumptions). – Gerry Myerson Feb 02 '18 at 08:09
  • If the polynomial only contains even powers the substitution $x^2=t$ will reduce the degree by two and make the integral elementary – Yuriy S Feb 02 '18 at 08:09
  • In relation, see https://en.wikipedia.org/wiki/Carlson_symmetric_form – Yuriy S Feb 02 '18 at 08:57
  • Here: https://projecteuclid.org/download/pdf_1/euclid.chmm/1428686948 on the 5th page, the statement under ART. 165: For the elliptic integrals, if n = 4, we have the integrals $I_0$, $I_1$, $I_2$, $H_1$; n = 3, there are the integrals$I_0$, $I_1$, $H_1$. In the first of these cases we shall see that $I_1$ reduces to elementary integrals ... – Shine Feb 02 '18 at 22:20
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    On page 186, the author reduces to the case where the quartic in the denominator is a biquadratic (that is, only even powers of $x$ appear), and in the case (as @Yuriy says) you get something elementary. – Gerry Myerson Feb 03 '18 at 12:02
  • I see, thanks everyone! – Shine Feb 03 '18 at 21:56

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