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