If
$
\int \left( \frac{1}{x} + \frac{1}{x^3} \right) \left( \sqrt[23]{3x^{-24}} + x^{-26} \right) \, dx
$
is equal to
$
-\frac{\alpha}{3(\alpha + 1)} \left( 3x^\beta + x^\gamma \right)^{\alpha + 1} + C, \quad x>0,
$
where $ \alpha, \beta, \gamma \in \mathbb{Z} $ and $ C $ is the constant of integration, then $ \alpha + \beta + \gamma $ is equal to _______.