\begin{equation*}
\begin{split}
& u = x^2 + 5 \\\\
& du = 2x \: dx \rightarrow \: dx = \frac{du}{2x} \\\\
& \int \frac{x^3}{x^2+5} \: dx \\\\
& \int\frac{x^3}{x^2+5} \: \frac{du}{2 x} \\\\
& \frac{1}{2}\int\frac{x^2}{x^2+5} \: du \\\\
& \frac{1}{2}\int\frac{u-5}{u} \: du \\\\
& \frac{1}{2}\int 1-\frac{5}{u} \: du \\\\
& \frac{1}{2}(u-5\ln|u|) + C \\\\
& \frac{1}{2}(x^2+5-5\ln|x^2+5|) + C
\end{split}
\end{equation*}
