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 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)

I'm not too sure my question is suitable here, but I did not have any luck at math.stackexchange (https://math.stackexchange.com/questions/234989/int-infty-0xr-s-11-x-s1-x2-fracrm2dx) so I thought I'd try and see if I can get some help here.