I would expand and integrate term by term.
Now putting x to the 4th power, and 4 in the denominator.
Not the Final Answer
However, that's the not the final answer. Some special manipulation needs to be done. That's because the du is larger than dx (of the original problem). It think some sort of division needs to be tried.
Let's try it this way:
Note: The du almost matches the dx (in the original problem). However, there is an extra that needs to be gotten rid of.
If the dx was then division by 2 would be the key. But in this problem, that isn't the case.
Ok, let's try to get rid of the by subtraction.
Now, simply integrate term by term using the power rule. I see you have edited your post...once you have integrated, you don't want to use the integration symbol, that is removed when you write the anti-derivative. So, remove the integration symbol, reduce any fractions and add the constant of integration.
Where the denominators are 1, I would omit them.
I looked at methods of substitution, but did not readily find a method that would work. There may be a way, but my inclination was to just expand and integrate.