Integral with conflicting solutions Here's the integral:
$$\int\frac{x e^{2x}}{(1+2x)^2}dx$$
Every method I try to use either hits a dead end or makes the problem more complicated. The only way I've managed to actually complete the integral is using integration by parts and distributing everything all the way out which but that ends up being a hugely complex mess of different terms that I have no way of verifying.
EDIT:
The answer in the back of the book and the answer given by Mathematica is:
$$\frac{e^{2x}}{8x+4}$$
However I can't seem to get there. I feel like there must be a way to arrange the completed integral algebraically so that all the complicated terms cancel and you end up with the nice clean answer the book gives, but after an hour of trial and error I haven't found it yet.
Edit: Apologies I did make a typo in this post, it has been corrected but I didn't make that error when computing the problem.
To confirm (for you and myself):


 A: THIS FIRST SECTION APPLIED TO THE ORIGINAL QUESTION, BEFORE EDITING
The answer in the back of your book is wrong, unless you mistyped something here. Here is a graph of the derivatives of both functions. You can see that they are not equal. They don't even have the same singularities.

If you change the question to
$$\int\frac{x e^{2x}}{(2x+1)^2}\,dx$$
then the answer in the back of the book is correct. Do you need help with that problem?

HELP FOR THE REVISED QUESTION
Apparently you do want help with the corrected problem. When you are stuck on doing an integration and you know the correct answer, a clue is to write out taking the derivative of the answer and getting the integrand of your question. You can then do that all backward.
If that makes the final answer seem too "magic," as it does here, replace the reverse-of-the-product-rule with an integration by parts.
Here is the method the book probably wants you to use: Do integration by parts, with $e^{2x}$ as the $u$ and the rest as $dv$.
Do you need more help?
