Find the second order derivatives of the function.

Let $y=x^{3} \log x$

Then,

$\frac{d y}{d x}=\frac{d}{d x}\left[x^{3} \log x\right]=\log x \cdot \frac{d}{d x}\left(x^{3}\right)+x^{3} \cdot \frac{d}{d x}(\log x)$

$=\log x \cdot 3 x^{2}+x^{3} \cdot \frac{1}{x}=\log x \cdot 3 x^{2}+x^{2}$

$=x^{2}(1+3 \log x)$

$\therefore \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left[x^{2}(1+3 \log x)\right]$

$=(1+3 \log x) \cdot \frac{d}{d x}\left(x^{2}\right)+x^{2} \frac{d}{d x}(1+3 \log x)$

$=(1+3 \log x) \cdot 2 x+x^{2} \cdot \frac{3}{x}$

$=2 x+6 x \log x+3 x$

$=5 x+6 x \log x$

$=x(5+6 \log x)$