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In this paper, we address the integration problem of the isomonodromic system of quantum differential equations ($qDE$s) associated with the quantum cohomology of $\mathbb P^1$-bundles on Fano varieties. It is shown that bases of solutions of the $qDE$ of the total space of the $\mathbb P^1$-bundle can be reconstructed from the datum of bases of solutions of the corresponding $qDE$ associated with the base space. This represents a quantum analog of the classical Leray-Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in arXiv:2005.08262, called $Borel$ $(α,β)$-$multitransf\!orms$. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer-Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen $\mathbb P^1$-bundle. When applied to projective bundles on products of projective spaces, our results give Mellin-Barnes integral representations of solutions of $qDE$s. As an example, we show how to integrate the $qDE$ of blow-up of $\mathbb P^2$ at one point via Borel multitransforms of solutions of the $qDE$ of $\mathbb P^1$.