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In this paper, we address the integration problem of the isomonodromic systemof quantum differential equations ($qDE$s) associated with the quantumcohomology of $\mathbb P^1$-bundles on Fano varieties. It is shown that basesof solutions of the $qDE$ of the total space of the $\mathbb P^1$-bundle can bereconstructed from the datum of bases of solutions of the corresponding $qDE$associated with the base space. This represents a quantum analog of theclassical Leray-Hirsch theorem in the context of the isomonodromic approach toquantum cohomology. The reconstruction procedure of the solutions can beperformed in terms of some integral transforms, introduced in arXiv:2005.08262,called $Borel$ $(\alpha,\beta)$-$multitransf\!orms$. We emphasize theemergence, in the explicit integral formulas, of an interesting sequence ofspecial functions (closely related to iterated partial derivatives of theB\"ohmer-Tricomi incomplete Gamma function) as integral kernels. Remarkably,these integral kernels have a universal feature, being independent of thespecifically chosen $\mathbb P^1$-bundle. When applied to projective bundles onproducts of projective spaces, our results give Mellin-Barnes integralrepresentations of solutions of $qDE$s. As an example, we show how to integratethe $qDE$ of blow-up of $\mathbb P^2$ at one point via Borel multitransforms ofsolutions of the $qDE$ of $\mathbb P^1$.

Borel $(\alpha,\beta)$-multitransforms and Quantum Leray-Hirsch:  integral representations of solutions of quantum differential equations for  $\mathbb P^1$-bundles