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  • $\begingroup$ RootSum[...] is a reasonable answer, see e.g. a related question How do I work with Root objects? $\endgroup$ Commented Dec 29, 2014 at 19:03
  • $\begingroup$ @ Artes thanks for your comment, I am sorry that I cant understand the logic of root sum from the link you mentioned above, could you please briefly describe it here? $\endgroup$ Commented Dec 29, 2014 at 19:11
  • $\begingroup$ @BushraMajeed You will not understand RootSum unless you learn about Root. You have to try a bit harder. You can compute that expression for special values of prarameters e.g. With[{Q = 1, m = 2, α = 0}, -RootSum[ Q^2 - 2 m #1 + #1^2 - α #1^3 &, (Log[r - #1] #1^2)/(2 m - 2 #1 + 3 α #1^2) &]]. $\endgroup$ Commented Dec 29, 2014 at 19:22
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    $\begingroup$ how about Integrate[r^2/(-\[Alpha] r^3 + r^2 - 2 m r + Q^2), r] //Normal//ToRadicals? $\endgroup$ Commented Dec 29, 2014 at 19:30
  • $\begingroup$ @chris I believe that makes a good answer, probably better if you explain a bit about RootSum[]. I'll surely upvote it. $\endgroup$ Commented Dec 29, 2014 at 20:22