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My problem is

For what values of $A$ and $B$ can this integral be expressed in terms of known or elementary functions?

$$\int \frac{1+(Ax)^{2B}}{x\sqrt{(x^{2B})-[1+(Ax)^{2B}]^{2}}} dx$$

I have tried integration by substitution of $u=1+(Ax)^{2B}$

and managed to get

$$\frac{1}{2BA^{B}}\int\frac{du}{-A^{2B}u^{2}+u-1} + \int \frac{dx}{x\sqrt{(x^{2B})-[1+(Ax)^{2B}]^{2}}}$$

the first term I believe I can integrate, the second however I am unsure about and I'm not sure if this is how I should be tackling this problem in first place.

Any advice or help would be much appreciated.

Thank you.

Vinyl_cape_jawa
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user134590
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    It may help to use $$x^{2B}-[1+(Ax)^{2B}]^2=[x^B+1+(Ax)^{2B}]\cdot [x^B-1-(Ax)^{2B}]$$ – clathratus Feb 25 '19 at 23:28
  • With the help of clathratus simplification and Mathematica it look like that there is a general complex solution with elliptic functions for $A \in n $ and any B – stocha Feb 26 '19 at 14:55

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