While working on a problem I came across the following interesting result.
Let: $$ H_{nm}(x)=\int_0^\infty t^{x-1}e^{-t}\log t\;F(-n;x;t)F(-m;x;t)\;dt, $$ where $n,m$ are non-negative integer numbers, $x$ is positive real number and $$F(a;b;t)=\sum_{k\ge0}\frac{a^{\overline k}}{b^{\overline k}}\frac{t^k}{k!}$$ is the confluent hypergeometric function.
Since the integral is symmetric with respect to permutation of $n$ and $m$ in what follows $n\ge m$ is assumed.
By numerical evidence the integral evaluates to the following simple values: $$ H_{nm}(x)=\frac{n!\;\Gamma^2(x)}{\Gamma(x+n)}\times \begin{cases} \displaystyle\psi(x+n),& n=m; \\ \displaystyle\frac1{m-n},&n\ne m. \end{cases}\tag1 $$ where $\Gamma(x)$ and $\psi(x)$ are the gamma and digamma functions, respectively.
Is there a simple way to prove the relations $(1)$?