Example 9. Find $\int \frac{\sqrt[6]{5-3 \sqrt[3]{x}}}{\sqrt[3]{x^{2}}} d x$.
Solution. We have $m=-\frac{2}{3}, n=\frac{1}{3}, p=\frac{1}{6}$ and $\frac{m+1}{n}=-2-$ an integer, so we make the second Chebyshev substitution $5-3 \sqrt[3]{x}=t^{6}$. Therefore, $x^{1 / 3}=\frac{1}{3}\left(5-t^{6}\right), x^{2 / 3}=\frac{1}{9}\left(5-t^{6}\right)^{2}, x=\frac{1}{27}\left(5-t^{6}\right)^{3}$, $d x=-\frac{2}{3} t^{5}\left(5-t^{6}\right)^{2} d t$. After the substitutions, we get:

$$
\begin{aligned}
& \int \frac{\sqrt[6]{5-3 \sqrt[3]{x}}}{\sqrt[3]{x^{2}}} d x=-\frac{2}{3} \int \frac{t \cdot t^{5}\left(5-t^{6}\right)^{2}}{\frac{1}{9}\left(5-t^{6}\right)^{2}} d t=-6 \int t^{6} d t= \\
& \quad=-\frac{6}{7} t^{7}+C=-\frac{6}{7}(5-3 \sqrt[3]{x})^{7 / 6}+C=-\frac{6}{7} \sqrt[6]{(5-3 \sqrt[3]{x})^{7}}+C
\end{aligned}
$$

## 7.4. Integration of Various Combinations of Rational Functions

Integrals of the form $\int R\left(x,\left(\frac{a x+b}{c x+d}\right)^{m_{1} / n_{1}}, \ldots,\left(\frac{a x+b}{c x+d}\right)^{m_{k} / n_{k}}\right) d x$ can be reduced to integrals of rational functions by the substitution $\frac{a x+b}{c x+d}=$ $=t^{n}$, where $n-$ is the least common multiple (LCM) of the fractions $\frac{m_{1}}{n_{1}}, \ldots, \frac{m_{k}}{n_{k}}$.

## Examples with Solutions