<s>[INST] Having trouble solving $\int x(2x+5)^8dx$. I am not sure how to solve $\int x(2x+5)^8dx$.
I have tried some different things, but nothing seems to give me the cancelling effect I need to solve the integral.
My Process:
Let $u=2x+5$
Then $du = 2dx$ and $dx = \frac{1}{2}du$
so $\int x u^8 \frac{1}{2} \;du = \frac{1}{2}\int x u^8 \; du$
This is where I get stuck. 
The answer should be $\frac{1}{40}(2x+5)^{10}-\frac{5}{36}(2x+5)^{9} + C$ according to my textbook.
How can I solve this problem?
 [/INST] Let $\displaystyle I=\int (2x+5)(2x+5)^8\,dx=\int (2x+5)^9\,dx=\frac{1}{20}(2x+5)^{10}$
And let $J=\displaystyle\int(2x+5)^8
\,dx=\frac{1}{18}(2x+5)^9$
I exclude constants until the end.
Then, $\displaystyle \int x(2x+5)^8\,dx=\frac{1}{2}(I-5J)+C=\frac{1}{40}(2x+5)^{10}-\frac{5}{36}(2x+5)^9+C$.
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