# $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral

$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$,

where $s$, $r$ and $m$>1 are positive integers.

My question is whether a closed form solution for this integral exists. By closed form here I mean an expression in terms of a finite number of special functions. There is some hope for a closed form solution I think given the output below from symbolic software, but I have been unable to find a formula yet.

To give a bit of context the integral is the $s$-th moment of the real random variable $x/(1+x)$, $x>0$, when the probability density function of $x$ is, up to a constant that I have omitted for simplicity, is $x^{r - 1}(1 + x^2)^{-\frac{rm}{2}}$. All these moments should exist. I am trying to understand whether they are expressible in terms of a finite number of special functions.

I was not able to get help from Mathematica or Maple. Both Mathematica and Maple give me a solution containing a $\Gamma$ function evaluated at negative integers. More precisely, if I input in Mathematica

Integrate[ x^(r + s - 1) (1 + x)^-s (1 + x^2)^(-r m/2), {x, 0, \[Infinity]}, Assumptions -> {r \[Element] Integers, s \[Element] Integers, m \[Element] Integers, r > 0, s > 0, m > 1}]

I obtain

(1/Gamma[s]) Gamma[(-1 + m) r] Gamma[ r - m r + s] HypergeometricPFQ[{(m r)/2, -(r/2) + (m r)/2, 1/2 - r/2 + (m r)/2}, {1/2 - r/2 + (m r)/2 - s/2, 1 - r/2 + (m r)/2 - s/2}, -1] + (1/( 2 Gamma[(m r)/ 2]))(-s Gamma[1/2 (-1 + (-1 + m) r - s)] Gamma[ 1/2 (1 + r + s)] HypergeometricPFQ[{1/2 + s/2, 1 + s/2, 1/2 + r/2 + s/2}, {3/2, 3/2 + r/2 - (m r)/2 + s/2}, -1] + Gamma[1/2 ((-1 + m) r - s)] Gamma[(r + s)/ 2] HypergeometricPFQ[{1/2 + s/2, r/2 + s/2, s/2}, {1/2, 1 + r/2 - (m r)/2 + s/2}, -1])

which is:

$\frac{\Gamma(r(m-1))\Gamma(s-r(m-1))}{\Gamma(s)} {}_{3}F_{2}\left( \frac {mr}{2},\frac{r(m-1)}{2},\frac{1+r(m-1)}{2};\frac{1-s+r(m-1)}{2},1+\frac {s-r(m+1)}{2};-1\right) -\frac{s\Gamma\left(\frac{r(m-1)-1-s}{2}\right)\Gamma\left( \frac{1+r+s}{2}\right) } {2\Gamma(\frac{mr}{2})}{}_{3}F_{2}\left( \frac{1+s}{2},1+\frac{s}{2} ,\frac{1+r+s}{2};\frac{3}{2},\frac{3+s-r(m-1)}{2};-1\right) +\Gamma\left( \frac{r(m-1)-s}{2}\right) \Gamma\left( \frac{r+s} {2}\right) {}_{3}F_{2}\left( \frac{1+s}{2},\frac{r+s}{2},\frac{s}{2} ;\frac{1}{2},1+\frac{s-r(m-1)}{2};-1\right)$

(same solution for Maple) As you can see, there is a $\Gamma$ function evaluated at $s-r(m-1)$ which can be a negative integer, so it seems to me that Mathematica and Maple are giving a solution that holds for most real values of $(r,m,s)$ but not necessarily for the values I'm interested in ($r,m,s$ are positive integers in my problem)

• Please give $\LaTeX$ format to the solution. As you suspect, the $\Gamma$ function has poles at negative integers, and it's likely that both MMA and Maple either are ignoring that the arguments are integers, or there are other terms canceling the poles. It's hard to say without a properly formatted solution. – Pragabhava Nov 11 '12 at 18:15
• @Pragabhava Edited, thanks – mark Nov 11 '12 at 22:38
• If I input Integrate[x^(r + s - 1) (1 + x)^-s (1 + x^2)^(-r m/2), {x, 0, \[Infinity]}, Assumptions -> {r \[Element] Integers, s \[Element] Integers, m \[Element] Integers}] to MMA, I get ConditionalExpression[...,r < m r && r + s > 0]. This means the solution is only valid when $r < m$ and $0 < r + s$. It's clear from the form of the original expression that the integral will not converge for all values of $m$, $r$ and $s$. – Pragabhava Nov 11 '12 at 22:50
• The condition in ConditionalExpression is $r < m r$ (not $r < m$) which is always satisfied because $r$ is a positive integer and $m$ is a positive integer larger than 1. – mark Nov 12 '12 at 0:30
• Why do you say that "it is clearer from the form of the original expression that the integral will not converge for all values of $m$, $r$ and $s$." I cannot see that. – mark Nov 12 '12 at 0:31