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Is there a closed form for the following integral (with $a$ and $b$ are given constants)?

$$\int \frac{dx}{x^2 + a x^b}$$

Trung Phan
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  • Welcome to Math.Stackexchange! What have you tried so far? – Thomas Davis Mar 07 '19 at 00:47
  • Sadly, i don't even know where to start. I did try Mathematica though, but didn't get an answer for the general a and b values. – Trung Phan Mar 07 '19 at 01:00
  • @TrungPhan - are there any constraints on $b$? –  Mar 07 '19 at 02:49
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    If you reposition the integral into the form:

    $$\frac{1}{a} \int_0^x \frac{t^{-2}}{\frac{1}{a} + t^{b - 2}} :dt $$

    And then you can form an Incomplete Beta Representation which I discuss in detail here:

    https://math.stackexchange.com/questions/3057298/solving-used-real-based-methods-int-0x-fractk-lefttn-a-rightm-d

     –  Mar 07 '19 at 02:51
  • @DavidG. This was a real good post ! – Claude Leibovici Mar 07 '19 at 06:19
  • @ClaudeLeibovici - it comes into handy A LOT! :-) –  Mar 07 '19 at 07:37

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