Can $x!H_x$ be written as an algebraic or exponential function?

Question

$H_x$ is given to be the Harmonic series $=\sum_{n=1}^x \frac{1}{n}$

$x!H_x$ is always an integer, when $x$ is an integer, because

$$H_x = \frac{x!}{x!}(1 + 1/2 + 1/3 + ... + 1/x) = \frac{x!+\frac{x!}{2}+\frac{x!}{3}+...+\frac{x!}{x}}{x!}$$

Can one write an algebraic or exponential function to model this set of integers? Or is $x!H_x$ the best closed form?

Answer 1

$$ \Gamma (1+ x) \, \left( \gamma + \psi (1+x) \right) = \gamma \, \Gamma (1+ x) + \Gamma' (1+ x) $$

https://en.wikipedia.org/wiki/Digamma_function#Relation_to_harmonic_numbers

https://en.wikipedia.org/wiki/Gamma_function