Integrate the function $\frac{x^{2}}{(2+3x^{3})^{3}}$.

Let $2+3x^{3} = t$.
Then,differentiating both sides with respect to $x$,we get $9x^{2} dx = dt$,which implies $x^{2} dx = \frac{1}{9} dt$.
Substituting these into the integral:
$\int \frac{x^{2}}{(2+3x^{3})^{3}} dx = \int \frac{1}{9t^{3}} dt = \frac{1}{9} \int t^{-3} dt$.
Using the power rule $\int t^{n} dt = \frac{t^{n+1}}{n+1} + C$:
$= \frac{1}{9} \left( \frac{t^{-2}}{-2} \right) + C = -\frac{1}{18t^{2}} + C$.
Substituting $t = 2+3x^{3}$ back into the expression:
$= -\frac{1}{18(2+3x^{3})^{2}} + C$,where $C$ is an arbitrary constant.