# Integration of an inverse polynomial

1. Jun 9, 2017

• Member advised that the homework template must be used
Hello,

I want to integrate this expression :

∫ (x5 + ax4 + bx3 + cx2 + dx)-1

between xmin>0 and xmax>0

a is positive but b, c and d can be positive or negative.

I have no idea to integrate this expression... Do you have methods to do this ?

2. Jun 9, 2017

### marcusl

You have not written a valid integral. You must reformulate your problem.

3. Jun 9, 2017

### Ray Vickson

You can solve the problem when you are given numerical values of $a,b,c,d$, by finding the roots of the polynomial using numerical methods, and so write the integrand in partial fractions. However, for general symbolic values of $a,b,c,d$ you are out of luck, because it is a rigorously-proven theorem that there is NO finite formula for the roots of a general polynomial of degree 5 or more. See, eg., https://en.wikipedia.org/wiki/Abel–Ruffini_theorem .

4. Jun 9, 2017

### marcusl

Sorry, I took dx to be the differential variable but I see now that you are using "d" as a constant, and that there must be an implied differential dx in your equation. Please ignore my post.

5. Jun 9, 2017