$$\int \frac{dx}{(ax^2 +b)\sqrt{px^2+q}}$$ A good substitution for this will be $x=\frac{1}{t}$ $$dx=-\frac{1}{t^2}$$ $$\int \frac{\frac{-1}{t^2}{dt}}{(\frac{a}{t^2} +b)\sqrt{\frac{p}{t^2}+q}}$$ $$\int \frac{-t dt }{(a +bt^2)\sqrt{p + qt^2}}$$ Then I have to go for one more substitution $$p+qt^2=z^2$$
I want to know is there any other good substitution which can solve it in single substitution .
If you let $$ u=\frac{x}{\sqrt{px^2+q}} $$ you will get $$ \int\frac{1}{b-(bp-aq)u^2}\,du, $$ which I think you can handle.