In this article, we present the exact expression of the L 2-norm of the forward stochastic integral driven by the multi-dimensional fractional Brownian motion with parameter 1 2 < H < 1. The class of integrands only requires rather weak integrability conditions compatible w.r.t. a random finite measure whose density is expressed as a second-order polynomial of the underlying driving Gaussian noise. A simple consequence of our results is the exact expression of the L 2-norm for the pathwise Young integral.