Example 8. Find $\int \sqrt{x}(5-3 \sqrt[3]{x})^{2} d x$.
Solution. We have $m=\frac{1}{2}, n=\frac{1}{3}, p=2$. Let's make the first Chebyshev substitution $x=t^{6}$, or $t=\sqrt[6]{x}$. From here $\sqrt{x}=t^{3}, 5-3 \sqrt[3]{x}=$ $=5-3 t^{2}, d x=6 t^{5}$ Substituting into the desired integral, we get:

$$
\begin{aligned}
& \int \sqrt{x}(5-3 \sqrt[3]{x})^{2} d x=\int t^{3}\left(5-3 t^{2}\right)^{2} \cdot 6 t^{5} d t= \\
& =6 \int\left(25 t^{8}-30 t^{10}+9 t^{12}\right) d t=\frac{50}{3} t^{9}-\frac{180}{11} t^{11}+\frac{54}{13} t^{13}+C= \\
& =\frac{50}{3} \cdot x \sqrt{x}-\frac{180}{11} x \sqrt[6]{x^{5}}+\frac{54}{13} x^{2} \sqrt[6]{x}+C .
\end{aligned}
$$