Answer
$\displaystyle \frac{1}{5}\ln|x^{5}+\sqrt{x^{10}-2}|+C$
Work Step by Step
With $\left[\begin{array}{ll}
u=x^{5} & \\
du=5x^{4}, & x^{4}dx=\dfrac{du}{5}
\end{array}\right]$ the integral becomes $I=\displaystyle \frac{1}{5}\int\frac{du}{\sqrt{u^{2}-2}}.$
Table of integrals:
$\color{blue}{43. \displaystyle \quad\int\frac{du}{\sqrt{u^{2}-\mathrm{a}^{2}}}=\ln|u+\sqrt{u^{2}-a^{2}}|+C }$
$(\mathrm{a}^{2}=2)$
$I=\displaystyle \frac{1}{5}\ln|u+\sqrt{u^{2}-2}|+C$
... bring back $x$...
$=\displaystyle \frac{1}{5}\ln|x^{5}+\sqrt{x^{10}-2}|+C$