  In recent years, the theory for Leibniz integral rule in the fractional sense
has not been able to get substantial development. As an urgent problem to be
solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type
differentiation operators with general fractional-order of $n-1 <\alpha \leq
n$, $n \in \mathbb{N}$ . A rule of fractional differentiation under integral
sign with general order is necessary and applicable tool for verification by
substitution for candidate solutions of inhomogeneous multi-term fractional
differential equations. We derive explicit analytical solutions of generalized
Bagley-Torvik equations in terms of recently defined bivariate Mittag-Leffler
type functions that based on fractional Green's function method and verified
solutions by substitution in accordance by applying the fractional Leibniz
integral rule. Furthermore, we study an oscillator equation as a special case
of differential equations with multi-orders via the Leibniz integral rule.
