Integrating $\sqrt{ax^{2}-a^{2}x}$

Question

How do I integrate the following function?

$$\int \left( \sqrt{ax^{2}-a^{2}x} \right) \, {\rm d} x$$

Answer 1

Substitute $$\sqrt{ax^2-a^2x}=t+\sqrt{a}x$$ if it is $$a>0$$ by squaring we get $$x=\frac{t^2}{a^2-2t\sqrt{a}}$$ and find $dx$ $$x=\frac{2 \sqrt{a} t^3-a^2 t^2}{a^4-4 a t^2}$$ and $$dx-\frac{2 t \left(a^{3/2}+t\right)}{\sqrt{a} \left(a^{3/2}+2 t\right)^2}dt$$ the result should be $$\frac{\sqrt{a x (x-a)} \left(a^2 \tan ^{-1}\left(\frac{\sqrt{x}}{\sqrt{a-x}}\right)+\sqrt{x} \sqrt{a-x} (2 x-a)\right)}{4 \sqrt{x} \sqrt{a-x}}+C$$

Answer 2

Hint: Let $x=a\cosh^2 t$ then $$ax^2-a^2x=ax(x-a)=a^3\cosh^2t\sinh^2 t=\dfrac{a^3}{4}\sinh^22t$$