How to solve this indefinite integral?

Question

$$\int{(24ab-c-dx)^{3/2}\over a^2-b^2x^2} dx $$ I have tried to solve this integral for a case when $$\ dx=constant$$ inside in a bracket but i am stuck that how to deal this generally.Thanks in advanced.

Answer 1

Hint:)

Let $\dfrac{2ab-c}{d}=A$, $\dfrac{a}{b}=B$ and then substituation $A-x=u^2$ $$ \int{(24ab-c-dx)^{3/2}\over a^2-b^2x^2} dx = \dfrac{d^\frac32}{b^2}\int\dfrac{(A-x)^\frac32}{B^2-x^2}dx = -2\dfrac{d^\frac32}{b^2}\int\dfrac{u^4}{(B+A-u^2)(B-A+u^2)}du $$