Given, $\int f(x) d x=\psi(x)$
Let $ I=\int x^{5} f\left(x^{3}\right) d x$
Put $ x^{3}=t$
$\Rightarrow x^{2} d x=\frac{d t}{3} .....$(i)
$\therefore I=\frac{1}{3} \int t f(t) d t$
$=\frac{1}{3}\left[t \cdot \int f(t) d t-\int\left\{\frac{d}{d t}(t) \int f(t) d t\right\} d t\right]$
[integration by parts]
$=\frac{1}{3}\left[t \psi(t)-\int \psi(t) d t\right]$
$=\frac{1}{3}\left[x^{3} \psi\left(x^{3}\right)-3 \int x^{2} \psi\left(x^{3}\right) d x\right]+c$[from Eq. (i)]
$=\frac{1}{3} x^{3} \psi\left(x^{3}\right)-\int x^{2} \psi\left(x^{3}\right) d x+c$