If $\int\left(\frac{1}{x}+\frac{1}{x^3}\right)\left(\sqrt[7]{3 x^{-24}+x^{-26}}\right) \mathrm{d} x=-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^7\right)^{\frac{\alpha+4}{\alpha}}+C, x>0,(\alpha, \beta, \gamma \in \mathbf{Z})$, where C is the constant of integration, then $\alpha+\beta+\gamma$ is equal to $\_\_\_\_$ $\dot{\sim}$