Question

Solve $\int \frac{1}{\sqrt{(4-3x^2)}}+\frac{1}{\sqrt{[3-(x^2/4\left)\right]}}+\frac{1}{(5+4x^2)}+\frac{1}{5+(2-3x{)}^2}dx$

Answer 1

$\int \frac{1}{\sqrt{[2^2-{\left(\sqrt{3x}\right)}^2]}}+\frac{1}{\sqrt{[{\left(\sqrt{3}\right)}^2-{\left(x/2\right)}^2]}}$
$+\frac{1}{{\left(\sqrt{5}\right)}^2+{\left(2x\right)}^2}+\frac{1}{\sqrt{[{\left(\sqrt{5}\right)}^2+{(2-3x)}^2]}}dx$
$=\frac{1}{\sqrt{\left(3\right)}}{\sin }^{-1}\frac{x\sqrt{3}}{2}+2{\sin }^{-1}\frac{x}{2\sqrt{\left(3\right)}}$
$+\frac{1}{\sqrt{\left(5\right)}}.\frac{1}{2}{\tan }^{-1}\frac{2x}{\sqrt{\left(5\right)}}+\frac{1}{\sqrt{\left(5\right)}}(-\frac{1}{3}){\tan }^{-1}\frac{2-3x}{\sqrt{\left(5\right)}}$