Answer
$$\frac{x}{5 \sqrt{x^{2}+5}}+C$$
Work Step by Step
Given
$$ \int \frac{d x}{\left(x^{2}+5\right)^{3 / 2}}$$
Let
$$x=\sqrt{5}\tan u\ \ \ \ \ \ \ \ dx= \sqrt{5}\sec^2 udu $$
Then
\begin{align*}
\int \frac{d x}{\left(x^{2}+5\right)^{3 / 2}}&=\int \frac{\sqrt{5}\sec^2 udu}{\left(5\tan^{2}u+5\right)^{3 / 2}}\\
&= \int \frac{\sqrt{5}\sec^2 udu}{\left(5\sec^{2}u \right)^{3 / 2}}\\
&= \frac{1}{5}\int \cos udu\\
&=\frac{1}{5}\sin u+C\\
&= \frac{x}{5 \sqrt{x^{2}+5}}+C
\end{align*}
