Answer
$$\ln |x+5|-\frac{25}{2(x+5)^{2}}+\frac{10}{x+5}+c$$
Work Step by Step
Given $$ \int \frac{x^2d x}{(x+5)^{3}}$$
Let
$$u=x+5 \ \ \ \Rightarrow \ \ du= dx $$
then
\begin{align*}
\int \frac{x^2d x}{(x+5)^{3}} &=\int \frac{(u-5)^2d u}{u^{3}} \\
&=\int \frac{(u^2-10u+25)d u}{u^{3}} \\
&= \int \frac{du}{u} -10\int \frac{du}{u^2}+25\int \frac{du}{u^3}\\
&=\ln |u| +\frac{10}{u}-\frac{25}{2u^2}+c\\
&=\ln |x+5|-\frac{25}{2(x+5)^{2}}+\frac{10}{x+5}+c
\end{align*}
