Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ?

A geometric Brownian motion $X=(X_t)_{t \geq 0}$ satisifies $dX_t = \sigma X_t \, dW_t$ where $W=(W_t)_{t \geq 0}$ denotes a Brownian motion and the associated integrated Brownian motion is $\int_0^t X_s \, ds$. The Laplace transform of an integrated gometric Brownian motion is thus

$$ \mathcal{L}(\lambda) = \mathbb{E}\left[e^{-\lambda \int_0^t X_s ds } \right]$$