Example 7. Find $\int\left(\frac{3}{\sqrt{x}}+1\right)^{2} \cdot x d x$.
Solution. In the integrand, we will square, expand the brackets, and apply the linearity property. We sequentially obtain

$$
\begin{aligned}
& \int\left(\frac{3}{\sqrt{x}}+1\right)^{2} \cdot x d x=\int\left(\frac{9}{x}+\frac{6}{\sqrt{x}}+1\right) x d x= \\
& =9 \int d x+6 \int \sqrt{x} d x+\int x d x= \\
& =\left\{\begin{aligned}
& \int \sqrt{x} d x=\int x^{1 / 2} d x=\frac{2}{3} x^{3 / 2}+C=\frac{2}{3} \sqrt{x^{3}}+C \\
& \text { division by a fraction is equivalent to multiplying by its } \\
& \text { reciprocal; when integrating radicals, the answer should } \\
& \text { also be expressed in radicals }
\end{aligned}\right\}= \\
& =9 x+4 \sqrt{x^{3}}+\frac{x^{2}}{2}+C .
\end{aligned}
$$