I would like to solve the following integral $$\int \frac{dx}{x^2 \sqrt[4]{(a-x^2)(b+x^2)}},$$ where $a$ and $b$ are the real constants.
My attempt: $$\sqrt[4]{(a-x^2)(b+x^2)} = \sqrt[4]{-\bigg[x^4 + (b-a)x^2 + (\frac{a-b}{2})^2\bigg]+(\frac{a+b}{2})^2} = \sqrt[4]{(\frac{a+b}{2})^2 - (x^2-\frac{a-b}{2})^2}$$
Substituting $t=x^2 - \frac{a-b}{2}$ one gets $$\int \frac{dt}{2(t+ \frac{a-b}{2})^\frac{3}{2}\sqrt[4]{(\frac{a+b}{2})^2 - t^2}}$$ I do not know what to do here!