I would like to integrate: $\int_0^\infty (ax^2 + bx + c)^n dx$, where $n$ is a negative integer. Does anyone know a method to do this?

My obvious first try has been using Mathematica, but it doesn't find a solution. And, I've not found it by going through Abramowitz and Stegun (although I may have missed a similar solution).

  • $\begingroup$ A change of variables could go in one of two ways... either as the solutions of $ax^2+bx+c$, or as $u=ax^2+bx+c$, $du=2axdx+bdx$ and replace. Of course, the integration over $[0, \infty)$ of a quadratic (for positive $n$) will always be $\pm\infty$, so the actual integration may be moot... $\endgroup$
    – abiessu
    Sep 16, 2013 at 13:50
  • $\begingroup$ It will exist if $n<-1/2$. Partial fractions may help if $n$ is an integer. $\endgroup$
    – Empy2
    Sep 16, 2013 at 13:52
  • $\begingroup$ @Michael: true, but I'm assuming that you mean $n \le -1$ given that $n\in \mathbb Z-\{0\}$. $\endgroup$
    – abiessu
    Sep 16, 2013 at 13:54
  • $\begingroup$ In fact in my real problem it will always be the case that $n < 1$. $\endgroup$ Sep 16, 2013 at 13:56
  • $\begingroup$ Now we're getting somewhere! Just put that info plus what you have tried in your question statement and we'll be able to help you properly. :-) $\endgroup$
    – abiessu
    Sep 16, 2013 at 13:57

1 Answer 1


Use simple fractions: $$ \frac{1}{(ax^2+bx+c)^n}=\frac{1}{a^n}\frac{1}{(x-x_1)^n(x-x_2)^n}=\frac{1}{a^n}\sum_{k=1}^n\frac{A_k}{(x-x_1)^k} + \frac{B_k}{(x-x_2)^k} $$


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