A note on the Irrationality of $ζ(5)$ and Higher Odd Zeta Values

For $n\in\mathbb{N}$ we define a double integral \begin{equation*} I_n=\frac{1}{24}\int_0^1\int_0^1 \frac{-\log^3(xy)}{1+xy} (xy(1-xy))^n \ dxdy\end{equation*} We denote $d_n=\text{lcm}(1,2,...,n)$ and prove that for all $n\in\mathbb{N}$, \begin{equation*} d_n I_n= 15 (-1)^n 2^{n-4} d_n \zeta(5)+(-1)^{n+1} d_n \sum_{r=0}^{n} \binom{n}{r}\left(\sum_{k=1}^{n+r} \frac{(-1)^{k-1}}{k^5}\right) \end{equation*} Now if $\zeta(5)$ is rational, then $\zeta(5)=a/b$, $(a,b)=1$ and $a,b\in\mathbb{N}$. Then we take $n\geq b$ so that $b|d_n$. We show that for all $n\geq 1$, $0