$\begin{aligned} & \text{If } 5(f(x))^2 = x f(x) + 30 \text{ and } \\ & \int \frac{3 x^3 + (1 - 30 x^2) f(x)}{(10 f(x) - x)(x^3 - f(x))^2} dx \\ & = \frac{A}{B x^3 + D f(x)} + C, \text{ then } A + B + D = \end{aligned}$

If $\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan x-\sin \theta}+B \sqrt{\cos \theta-\cot x \sin \theta}+C,$ where $C$ is the integration constant,then $AB$ is equal to

Let $I_{n}(x)=\int_{0}^{x} \frac{1}{(t^{2}+5)^{n}} dt, n=1, 2, 3, \ldots$. Then

Let $I_n = \int \tan^n x dx, (n > 1)$. If $I_4 + I_6 = a \tan^5 x + b x^5 + C$,where $C$ is the constant of integration,then the ordered pair $(a, b)$ is equal to:

$\int \frac{3x+2}{4x^2+4x+5} dx = A \log(4x^2+4x+5) + B \tan^{-1}\left(\frac{2x+1}{2}\right) + c$,then $A+B=$

$\int \sqrt{x^2-8 x+7} \, dx = $ . . . . . . $+ C$.