Can $\int_{0}^{\infty}\frac{dx}{\sqrt[\leftroot{-1}\uproot{1}n]{x(x+a^n)(x+1)^{n-1}}}$ be expressed with an elliptic integral?

Asked Jun 11 '17 at 06:01

Active Jun 11 '17 at 06:01

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This integral came up earlier and I'm hoping to express it with elliptic integrals -- or any interesting functions really $$I_n=\int_{0}^{\infty}\frac{dx}{\sqrt[\leftroot{-1}\uproot{1}n]{x(x+a^n)(x+1)^{n-1}}}$$

It seems possible for $n$ in general only because my source -- i.e. wolfram alpha -- gives some interesting expressions for other $n=2,3,4$. Ex. $$I_2=2K(1-a^2)$$ $$I_4=2\sqrt{2}K\Big(\frac{1-a^2}{2}\Big)$$ where $K$ is the complete elliptic integral of the first kind.

Does anyone know how exactly these identies are derived in the first place? I'd also appreciate any good introductory resources about elliptic integrals.

This post is related and has the info that got me started: To evaluate $\int_0^{+\infty} \frac{\;\mathrm dx}{\sqrt[3]{x^3+a^3}\sqrt[3]{x^3+b^3}\sqrt[3]{x^3+c^3}}$

asked Jun 11 '17 at 06:01

I got a solution with Mathematica – Dr. Sonnhard Graubner Jun 11 '17 at 07:14
Did you get anything interesting beyound $I_2$ and $I_4$. I have the strange feeling that for any other value of $n$, the result would involve hypergeometric functions. – Claude Leibovici Jun 11 '17 at 07:32
@DrSonnhardGraubner What did it give you? – Antonio DJC Jun 11 '17 at 07:59
@ClaudeLeibovici $I_3$ also had an expression -- though very complicated -- in terms of elliptic integrals. – Jun 11 '17 at 20:17
@ChristianWoll What assumptions are we allowed to make about the parameters $a$ and $n$? – David H Jun 18 '17 at 03:05
@DavidH $n$ is an integer. $a$ can pretty much be whatever gets results. I've been applying these solutions to matrices. I think conventionally, $a$ is complex. – Jun 18 '17 at 17:41

Can $\int_{0}^{\infty}\frac{dx}{\sqrt[\leftroot{-1}\uproot{1}n]{x(x+a^n)(x+1)^{n-1}}}$ be expressed with an elliptic integral?

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