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I need help with the integral.

\begin{equation} I_{n,m}(a,b)=\int dx \frac{x^n \ln(a x^2+b x)}{(a x^2+b x)^m}\,,\enspace\enspace \begin{array}{rcl}n,m &=& {1,2,3,\ldots},\enspace (n\geq m)\\ a, b&\neq&0 \end{array} \end{equation}

I would like to get a result that is some kind of terminating series involving a polynomial of $x$ and their logarithms.

I started by splitting the logarithm and factoring out an $x$ from the denominator:

\begin{equation} I_{n,m}(a,b) = \int dx \frac{x^{n-m}\ln(x)}{(a x+b)^m} + \int dx \frac{x^{n-m}\ln(a x+ b)}{(a x+b)^m} \end{equation}

But, now I'm stuck. I don't know how to calculate either of these integrals.

QuantumDot
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  • Have you tried to integrate simple cases (like $m, n = 1, 2$)? – rubik Dec 09 '15 at 14:51
  • @rubik ooh.. I think it is best to make a substitution: $y=(ax+b)$.. still working on it... – QuantumDot Dec 09 '15 at 17:22
  • @rubik Ok, I just realized that this integral is not suitable as a stackexchange question since it is just a laborious task with not too much insight. Can you initiate a vote to close? – QuantumDot Dec 10 '15 at 22:10
  • Well I think that it's an interesting problem, and you could just answer with a pointer to the solution (just a sketch of it). You can accept your own answer and it's encouraged too. – rubik Dec 11 '15 at 01:17

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