Mathematics Question from JEE Main 2025 · 7 April, Shift 2

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