Integral $\int {\left(b\left(\frac{1}{x}-\frac{1}{a}\right)+\frac{c}{2}\left(\frac{1}{a^2}-\frac{1}{x^2}\right)\right)}^{-\frac{1}{2}} dx$

Question

Evaluate: $$\int {\left(b\left(\frac{1}{x}-\frac{1}{a}\right)+\frac{c}{2}\left(\frac{1}{a^2}-\frac{1}{x^2}\right)\right)}^{-\frac{1}{2}} dx$$ I came to this integral while solving a problem on rectilinear motion.I failed to solve it.Please help.

Answer 1

Hint: Write your integral in the form

$$\pm\int \frac{\sqrt{2}ax}{\sqrt{x^2(c-2ab)+2a^2bx-2a^2c}}dx$$ and use the so-called Euler substitution. See here https://en.wikipedia.org/wiki/Euler_substitution