How to calculate the following expression:

$$\int_0^u{\ln(x)x^{k-1}e^{-x}}\;dx$$

As I know,

$$\int_0^\infty{\ln(x)x^{k-1}e^{-x}}dx = \Gamma(k)\Psi(k)$$

Are there any way to transfer the integral as expression by digamma, gammaln functions?

Or, are there any fast but also precise enough way to evaluate the integral? Thanks. Matlab's numerical integration directly is not so fast, but if there is a relation with the built in functions such as psi, gammaln, then that'll meet the need.