Does this Laplace transform have a closed form?

Question

I want to know if the Laplace transform of

$$x^\alpha (1+ax)^\beta$$

has any closed form?

I really appreciate your help.

Answer 1

Yes it does, but requires special functions known as Tricomi confluent hypergeometric function (see functions.wolfram.com). The integral reduces to the one stated in the linked page after change of variables $x = \frac{y}{a}$:

$$ LT_s(x^\alpha (1+a x)^\beta) = a^{\alpha - 1} \Gamma(1 + \alpha) U(1 + \alpha, 2 + \alpha + \beta, \frac{s}{a}) $$