Example 1.15. Find $\int \frac{d x}{(x+3)^{1 / 2}+(x+3)^{2 / 3}}$.
Solution. According to the given, the rationalizing substitution for the integral is $x+3=t^{6}$, since $\operatorname{LCM}(2,3)=6$. Then

$$
x=t^{6}-3, (x+3)^{1 / 2}=t^{3}, (x+3)^{2 / 3}=t^{4}, d x=6 t^{5} d t
$$

## We have

$$
\begin{gathered}
\int \frac{d x}{(x+3)^{1 / 2}+(x+3)^{2 / 3}}=6 \int \frac{t^{5} d t}{t^{3}(t+1)}= \\
=6 \int \frac{\left(t^{2}-1+1\right) d t}{t+1}=6 \int(t-1) d t+6 \int \frac{d t}{t+1}= \\
=3 t^{2}-6 t+6 \ln |t+1|+C= \\
=3 \sqrt[3]{x+3}-6 \sqrt[6]{x+3}=6 \ln |\sqrt[6]{x+3}+1|+C .
\end{gathered}
$$

Integrals of the form $\int R\left(x, \sqrt{a^{2}-x^{2}}\right) d x, \int R\left(x, \sqrt{a^{2}+x^{2}}\right) d x$, $\int R\left(x, \sqrt{x^{2}-a^{2}}\right) d x$ can be reduced to integrals of the form $\int R(\sin x, \cos x) d x$ using the following trigonometric or hyperbolic substitutions:

$$
x=a \sin t \text { or } x=a \tan t \text { or } x=a \sinh t, x=\frac{a}{\cos t} .
$$