I am trying to find the solution of the integral
\begin{align} I =\int_{0}^{\infty}e^{-t}t^{\alpha+1}\left[ _{1}F_{1}(-m; \alpha +1;t)\right]^{2}\log\left[ _{1}F_{1}(-m; \alpha +1;t)\right]^{2}dt \end{align}
where $m\in\mathbb{Z}_{>0}$ and $\alpha\in\mathbb{R}$.
My closest approach is thinking about the tabulated solution for \begin{align} I_{2}=\int_{0}^{\infty} t^{\alpha + 1}e^{-t}\left[_{1}F_{1}(-m; \alpha +1;t)\right]^{2q}dt=\Gamma(\alpha)F_{A}^{(2q)} \begin{pmatrix}, \alpha+2;-m,...,-m & \\ & ;1,...,1 \\ \alpha+1,...,\alpha+1 & \end{pmatrix}. \end{align}
So
\begin{align} I=\left.\frac{dI_{2}}{dq}\right\vert_{q=1}=\Gamma(\alpha)\frac{d}{dq}\left[F_{A}^{(2q)} \begin{pmatrix}, \alpha+2;-m,...,-m & \\ & ;1,...,1 \\ \alpha+1,...,\alpha+1 & \end{pmatrix}\right]_{q=1}, \end{align}
but because q is supposse to be an integer number, I am not sure if this has sense and if it has, I do not know how to calculate it.