Integration of an inverse polynomial

[Ado]

Hello,

I want to integrate this expression :

∫ (x⁵ + ax⁴ + bx³ + cx² + dx)^-1

between x_min>0 and x_max>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 ?

Thanks in advance !

 

[marcusl]

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

 

[Ray Vickson]

Hello,

I want to integrate this expression :

∫ (x⁵ + ax⁴ + bx³ + cx² + dx)^-1

between x_min>0 and x_max>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 ?

Thanks in advance !

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]

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.

 

[Ado]

Never mind! ;)
Thanks for your replies!