# Integration of an inverse polynomial

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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 ?

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

Ray Vickson
Homework Helper
Dearly Missed
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 ?

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 .

marcusl