I want to express the Inverse Laplace transform (arising from a fractional calculus problem) $$ F(t) = \mathcal{L}^{-1}\Big\{ \frac{1}{s}\frac{1}{1 + (k/s)^a + (l/s)^b}\Big\}(t) $$ in terms of special functions. Here $k$ and $l$ are some constants and $ a$ and $b$ are real numbers at least $1$. One can see that $F(t)$ generalizes the exponential function, since setting $a=b=1$ we have $F(t) = e^{-(k+l)t}$.
To make progress I made a Taylor expansion, and then a binomial expansion, then inverse transformed term by term, finding $$ F(t) = \sum_{n=0}^\infty [-(l t)^b]^n \sum_{m=0}^n \binom{n}{k} \frac{(k^a/l^b)^m t^{(a-b)m}}{\Gamma[(a-b)m + (1+bn)]}.$$ The second sum is reminiscent of the Mittag-Leffler function, defined as $$ E_{\alpha, \beta}(z) = \sum_{m=0}^\infty \frac{z^m}{\Gamma(\alpha m + \beta)},$$ although it has the wrong summation range and an extra binomial coefficient on each term.
Any suggestions would be welcome. Is there a way I can evaluate this inverse Laplace transform into special functions?