Let f(x)=∫ (d x)/((3+4 x^2) √(4-3 x^2)),|x|<2/(√ 3). If f(0)=0 and f(1)=1/(α β) tan^-1((α)/(β)), α, β>0, then α^2+β^2 is equal to ____.

Let $\displaystyle f(x)=\int \frac{d x}{\left(3+4 x^2\right) \sqrt{4-3 x^2}},|x|<\frac{2}{\sqrt{3}}$. If $\displaystyle f(0)=0$ and $\displaystyle f(1)=\frac{1}{\alpha \beta} \tan ^{-1}\left(\frac{\alpha}{\beta}\right)$, $\displaystyle \alpha, \beta>0$, then $\displaystyle \alpha^2+\beta^2$ is equal to $\displaystyle \_\_\_\_$.