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Math Help - An integral. Nothing too fancy... or not?

  1. #1
    Member Pranas's Avatar
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    An integral. Nothing too fancy... or not?

    Hi everyone.
    Sorry for such a lame thread, but I feel extremely rusty with my math...

    \ell  = 2 \cdot \int\limits_0^{ + \infty } {\frac{{\sqrt {{x^2} + 1} }}{{{{(x + 1)}^3}}}} dx

    Can you give a hint how to make this one nice and easy?

    P.S. I don't think it's important, but as a matter of fact it arose from evaluating the length of a curve \sqrt x  + \sqrt y  = 1

    Thanks!
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  2. #2
    A Plied Mathematician
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    Re: An integral. Nothing too fancy... or not?

    I don't think that integral is easy. Mathematica yields

    2\int_{0}^{\infty}\frac{\sqrt{x^{2}+1}}{(x+1)^{3}}  \,dx=-\frac{\text{MeijerG}[\{\{\frac{1}{2},1,2\},\{\}\},\{\{\frac{1}{2},\frac  {3}{2},2\},\{\}\},1]}{\pi^{3/2}},

    where

    \text{MeijerG}[\{\{a_{1},\dots,a_{n}\},\{a_{n+1},\dots,a_{p}\}\},  \{\{b_{1},\dots,b_{m}\},\{b_{m+1},\dots,b_{q}\}\},  z] is the MeijerG function

    G_{p\;q}^{m\,n}}\left(z\Bigg|\begin{matrix}a_{1}, \dots,a_{p}\\ a_{1},\dots,b_{q}\end{matrix}\right).

    Not very pretty, if you ask me.
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