I am wondering if the following expression can be processed a bit analytically, $$ E \left< e^{aX} \int_0^X e^{bu}dW(u)\right>, $$ where $W_u$ is the normal Brownian motion (1D Wiener process), and $X$ is a random variable.

I know that $E \left< \int_0^T e^{bu}dW(u)\right>$ is zero, so $E \left< \int_0^X e^{bu}dW(u)\right>$ should also be zero. However $e^{aX}$ and $\int_0^X e^{bu}dW(u)$ are clearly correlated, so the expectation of their product is not trivial.