$\begin{align}  & \int{{{(3u-5)}^{\tfrac{5}{2}}}}du \\ & \text{Using change of variable method} \\ & \text{Let }x=3u-5,\text{ }\frac{dx}{du}=3,\text{  }du=\frac{dx}{3} \\ & \therefore \int{{{(3u-5)}^{\tfrac{5}{2}}}}du=\int{{{x}^{\tfrac{5}{2}}}\tfrac{dx}{3}}=\frac{1}{3}\int{{{x}^{\tfrac{5}{2}}}dx}=\frac{1}{3}\left( \frac{{{x}^{\tfrac{5}{2}+1}}}{\tfrac{5}{2}+1} \right)+C \\ & \int{{{(3u-5)}^{\tfrac{5}{2}}}}du=\frac{1}{3}\left( \frac{{{x}^{\tfrac{7}{2}}}}{\tfrac{7}{2}} \right)+C=\frac{2{{x}^{\tfrac{7}{2}}}}{21}+C \\ & \int{{{(3u-5)}^{\tfrac{5}{2}}}}du=\frac{2}{21}{{(3u-5)}^{\tfrac{7}{2}}}+C \\\end{align}$