5. The sum of all real roots of the equation $x+\frac{x}{\sqrt{x^{2}-1}}=\frac{35}{12}$ is . $\qquad$
5. $\frac{35}{12}$.

From $x+\frac{x}{\sqrt{x^{2}-1}}=\frac{35}{12}$, we know $x>1$.
Let $x=\frac{1}{\cos \theta}\left(\theta \in\left(0, \frac{\pi}{2}\right)\right)$. Then $\frac{1}{\cos \theta}+\frac{1}{\sin \theta}=\frac{35}{12}$.
Let $t=\cos \theta+\sin \theta$. Then $35 t^{2}-24 t-35=0 \Rightarrow t=\frac{7}{5}$ (negative root discarded) $\Rightarrow \cos \theta=\frac{3}{5}$ or $\frac{4}{5}$.
Therefore, the roots of the original equation are $\frac{5}{3}$ and $\frac{5}{4}$, and the sum of the roots is $\frac{35}{12}$.