Answer
$$\int x(2x+5)^8dx=\frac{(2x+5)^9(18x-5)}{360}+c$$
Work Step by Step
To solve the integral $\int x(2x+5)^8dx$ we will use substitution $t=2x+5$ which gives us $x=\frac{x-5}{2}$ and $2dx=dt\Rightarrow dx=\frac{1}{2}dt.$ Putting this into the integral we get:
$$\int x(2x+5)^8dx=\int\frac{t-5}{2}t^8\frac{dt}{2}=\frac{1}{4}\int t^9dt-\frac{5}{4}\int t^8dt=
\frac{1}{4}\frac{t^{10}}{10}-\frac{5}{4}\frac{t^9}{9}+c=\frac{t^9}{4}(\frac{t}{10}-\frac{5}{9})+c$$
where $c$ is arbitrary constant. Now we have to express solution in terms of $x$ by expressing $t$ in terms of $x$:
$$\int x(2x+5)^8dx=\frac{t^9}{4}(\frac{t}{10}-\frac{5}{9})+c=\frac{(2x+5)^9}{4}(\frac{2x+5}{10}-\frac{5}{9})+c=\frac{(2x+5)^9}{4}\cdot\frac{18x+45-50}{90}+c=\frac{(2x+5)^9(18x-5)}{360}+c$$
