Example 1.19. Find $\int x^{1 / 2}\left(3+2 x^{3 / 4}\right)^{1 / 2} d x$.
Solution. In the considered case $m=1 / 2, n=3 / 4$, $p=1 / 2, \frac{m+1}{n}=2$ - an integer.

Let $t=\sqrt{3+2 x^{3 / 4}}$. Then $t^{2}=3+2 x^{3 / 4}, x=$ $=\left(\frac{t^{2}-3}{2}\right)^{4 / 3}, d x=\left(\frac{1}{2}\right)^{4 / 3} \frac{8}{3}\left(t^{2}-3\right)^{1 / 3} t d t$. Expressing the integrand in terms of the variable, we have:

$$
\begin{aligned}
& \int x^{1 / 2}\left(3+2 x^{3 / 4}\right)^{1 / 2} d x= \\
& =\int\left(\frac{t^{2}-3}{2}\right) 6(2 / 3) t\left(\frac{1}{2}\right)^{4 / 3} \frac{8}{3}\left(t^{2}-3\right)^{1 / 3} t d t= \\
& =\frac{2}{3} \int\left(t^{4}-3 t^{2}\right) d t=\frac{2}{3} \frac{t^{5}}{5}-2 \frac{t^{3}}{3}+C= \\
& \quad=\frac{2}{15} \sqrt{\left(3+2 x^{3 / 4}\right)^{5}}-\frac{2}{3} \sqrt{\left(3+2 x^{3 / 4}\right)^{3}}+C
\end{aligned}
$$

3) The number $\frac{m+1}{n}+p$ is an integer. Then the integral $\int x^{m}\left(a+b x^{n}\right)^{p} d x$ is rationalized by the substitution $t=\sqrt[s]{b+\frac{a}{x^{n}}}$, where $s-$ is the denominator of the rational number $\boldsymbol{f}$.