As part of my Master thesis, I came across quite horrible integrals involving hypergeometric functions and hope someone here has insight in how to compute them explicitly.
Let $N\geq 2$ and $k\in (-N,0)$ and $F$ the hypergeometric function 2F1: https://functions.wolfram.com/07.23.21.0011.01. I would like to compute the integrals
$$A( r):=\int_0^r (1+z^2)^{-N-k/2}z^{N-1}F(-k/2,1- \frac{k+N}{2}; \frac{N}{2};(\frac{z}{r})^2) dz$$ and $$B( r ):=\int_r^{\infty} (1+z^2)^{-N-k/2}z^{N-1+k}F(-k/2,1-\frac{k+N}{2}; \frac{N}{2};(\frac{r}{z})^2) dz$$
The expressions involving $F$ are well-defined because $|\frac{z}{r}|<1$ if $z<r$ and $|\frac{r}{z}|<1$ if $z>r$.
There are plenty of questions about evaluating integrals involving hypergeometric functions, for instance: Integral involving hypergeometric function. But here the arguments in $F$ are a bit more general.
There are many useful formulas involving integrals with hypergeometric functions. Note that F is of the form $F(a,a-b+1/2,b+1/2,.)$. Formula 15.3.17 in https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf seems useful to simplify the integrand:
$$F(a,b;2b;z)=(\frac12+\frac12\sqrt{1-z})^{-2a}F(a,a-b+1/2;b+1/2;(\frac{1-\sqrt{1-z}}{1+\sqrt{1+z}})^2)$$ but it gives the wrong exponent for $(1+z^2)$.
Background: The question comes from verifying Euler-Lagrange equations/steady states of a Keller-Segel system, the integrals appear in Lemma 6 of the paper https://arxiv.org/abs/1905.07788.
I want to show that the function $\rho_{HLS}(r)=(\frac{1}{1+r^2})^{N+k/2}$ (as (slightly different) defined in equation (6.3) in https://arxiv.org/abs/1612.08225) which maximises the HLS inequality $$\int_{\mathbb{R}^n\times \mathbb{R}^n} \rho(x)\rho(y)|x-y|^k dxdy\leq C_{HLS}||\rho||_m^2$$ solves the Euler-Lagrange equations corresponding to the functional $$\mathcal{F}_2[\rho]_2=C_{HLS}||\rho||_m^2-\int_{\mathbb{R}^n\times \mathbb{R}^n} \rho(x)\rho(y)|x-y|^k dxdy,$$ that is $$2\left(\int \rho^m dx\right)^{\frac{2}{m}-1}\rho(x)^{m-1}-\rho(x)\ast |x|^k=0$$
Using Lemma 6 to rewrite $\rho(x)\ast |x|^k$ in polar coordinates using hypergeometric functions and with $||\rho_{HLS}||_m=(\frac{2^{1-N}\pi{\frac{N+1}{2}}}{\Gamma(\frac{N+1}{2})})^{n+k/2}$ (see Evaluate $\int_{\mathbb{R}^n} \left(\frac{\lambda}{\lambda^2+|x|^2}\right)^N dx$), I want to check that
$$2^{\frac{3N-5}{2}}\pi^{\frac{N-1}{2}}\frac{\Gamma(\frac{N-1}{2})}{\Gamma(N-1)}(r^{k}A( r )+r^{-2n} B( r) )=\frac{2^{\frac{1+2(1-n)(2n+k)}{k}}\pi^{\frac{(n+1)(2n+k)}{k}}} {\Gamma(\frac{N+1}{2})^{\frac{2(2n+k)}{k}}}(\frac{1}{1+r^2})^{-k/2}$$