We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho $-integral, introduced by Jarník and Kurzweil. Let $({\mathcal{S}}_{\rho}\left(E\right),\parallel \xb7\parallel )$ be the space of all strongly $\rho $-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\parallel \xb7\parallel $. We show that each element in the dual space of $({\mathcal{S}}_{\rho}\left(E\right),\parallel \xb7\parallel )$ can be represented as a strong $\rho $-integral. Consequently, we prove that $fg$ is strongly $\rho $-integrable on $E$ for each strongly $\rho $-integrable function $f$ if and only if $g$ is almost everywhere...