I want to know if the Laplace transform of
$x^\alpha (1+ax)^\beta$
has any closed form?
I really appreciate your help.
I want to know if the Laplace transform of
$x^\alpha (1+ax)^\beta$
has any closed form?
I really appreciate your help.
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}) $