\begin{aligned} & \left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5 \\ & =2\left\{{ }^5 C_0 \cdot x^5+{ }^5 C_2 \cdot x^3\left(x^3-1\right)+{ }^5 C_4 \cdot x\left(x^3-1\right)^2\right\} \\ & =2\left\{5 x^7+10 x^6+x^5-10 x^4-10 x^3+5 x\right\} \\ & \Rightarrow \alpha=10, \beta=2, \gamma=-20, \delta=10 \\ & \text { Now, } 10 \mathrm{u}+2 \mathrm{v}=18 \\ & -20 \mathrm{u}+10 \mathrm{v}=20 \\ & \Rightarrow \mathrm{u}=1, \mathrm{v}=4 \\ & \mathrm{u}+\mathrm{v}=5 \end{aligned}