4. Calculate

$$
\int\left(5 x^{5}+3 x^{3}\right) \sqrt{x^{3}+x} d x, \quad x>0
$$
Solution. We have

$$
\int\left(5 x^{5}+3 x^{3}\right) \sqrt{x^{3}+x} d x=\int\left(5 x^{4}+3 x^{2}\right) \sqrt{x^{5}+x^{3}} d x
$$

With the substitution $x^{5}+x^{3}=t$, we have $\left(5 x^{4}+3 x^{2}\right) d x=d t, \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ and the integral becomes

$$
\int \sqrt{t} d t
$$

from which

$$
\int\left(5 x^{5}+3 x^{3}\right) \sqrt{x^{3}+x} d x=\frac{2}{3}\left(x^{6}+x^{4}\right) \sqrt{x^{3}+x}+\mathcal{C}
$$

$.3 p$