Laplace transform of the product of two gammas

Suppose that $X$ and $Y$ are both gamma distributed with shapes $a,b$ and unit scales / unit rates. To fix ideas, X has Laplace transform given by:

$$L_X(t) = mathbb{E}(e^{-tX}) = (1 + t)^{-a}$$

How can we compute the Laplace transform of the independent product $Z=XY$, that is the integral:

$$L_{XY}(t) = mathbb{E}(e^{-tXY}) = mathbb{E}(L_Y(tX)) = frac{1}{Gamma(a)}intlimits_{0}^{+infty} (1+tx)^{-b} x^{a-1} e^{-x} partial x text{ ?}$$