$$
\begin{aligned}
& \text { Let, } I=\int_I^{(\log x)^3 x^4 d x} \\
& =(\log x)^3 \cdot \frac{x^5}{5}-\int \frac{x^5}{5} \cdot 3(\log x)^2 \cdot \frac{1}{x} d x \\
& =\frac{1}{5} x^5(\log x)^3-\frac{3}{5}\left[\int x^4(\log x)^2 d x\right] \\
& =\frac{1}{5} x^5(\log x)^3-\frac{3}{5}\left[\frac{x^5}{5}(\log x)^2-\int \frac{x^5}{5} \cdot 2(\log x) \cdot \frac{1}{x} d x\right]
\end{aligned}
$$


$$
\begin{aligned}
& =\frac{1}{5} x^5(\log x)^3-\frac{3}{25} x^5(\log x)^2+\frac{6}{25} \int x^4 \log x d x \\
& =\frac{1}{5} x^5(\log x)^3-\frac{3}{25} x^5(\log x)^2 \\
& \qquad+\frac{6}{25}\left[\frac{x^5}{5} \log x-\int \frac{x^5}{5} \cdot \frac{1}{x} d x\right] \\
& =\frac{1}{5} x^5(\log x)^3-\frac{3}{25} x^5(\log x)^2 \\
& \quad+\frac{6}{125} x^5 \log x-\frac{6}{625} x^5+c \\
& =\frac{x^5}{625}\left[\operatorname{li25p}-75 p^2+30 p-6\right]+c \\
& \text { where, } p=\log x
\end{aligned}
$$