Difficult integration problem (unique headline) So I found this difficult integral at Physicsforum (I'm quite sure you are familiar with this forum) and I've been at it for about an hour now and it's quite frustrating because I can't get the answer.

$$\int \frac{4x^5 -1}{x^5 + x +1} dx $$

I would really love it if someone could help me with this
Taking the question as it is, it is a rational function and therefore cen be integrated by partial fractions. Fist divide the denominator into the numerator to get a fraction where the numerator has a smaller degree. Integrating the quotient gives $4x$. Now the problem is to factor the denominator. However it does not have any rational roots. It is of odd degree so has at least one real root, so irrational quantities will be an essential part of the answer. The other question is how many factors are linear and how many complex. You can work this out by graphing the function (maxima and minima).
Each linear factor gives a logarithm and each quadratic a inverse tangent. So you should be able to determine the general form of the integral, but not the specific constants involved.
Taking the question as it is, it is a rational function and therefore cen be integrated by partial fractions. Fist divide the denominator into the numerator to get a fraction where the numerator has a smaller degree. Integrating the quotient gives $4x$. Now the problem is to factor the denominator. However it does not have any rational roots. It is of odd degree so has at least one real root, so irrational quantities will be an essential part of the answer. The other question is how many factors are linear and how many complex. You can work this out by graphing the function (maxima and minima).
Each linear factor gives a logarithm and each quadratic a inverse tangent. So you should be able to determine the general form of the integral, but not the specific constants involved.