5 once again

$\large \int_0^\infty \sin{x^{5}} \ dx = \frac 1a \left(\sqrt b-1\right) \Gamma \left(\frac 1c \right)$

The equation above holds true for positive integers $a$, $b$, and $c$. Submit $abc$.

Notation: $\Gamma(\cdot)$ denotes the gamma function.

Hint: Use $e^{ix}= \cos{x} + i\sin{x}$.