Mathematics - Indefinite Integration Question with Solution | TestHub

$\begin{array}{rl}& \int \frac{1}{\sqrt[5]{(x-1{)}^4(x+3{)}^6}}dx=A{\left(\frac{\alpha x-1}{\beta x+3}\right)}^B+C\\ & I=\int \frac{1}{(x-1{)}^{4/5}(x+3{)}^{6/5}}dx\\ & I=\int \frac{1}{{\left(\frac{x-1}{x+3}\right)}^{4/5}(x+3{)}^2}dx\\ & \left(\frac{x-1}{x+3}\right)=t\Rightarrow \frac{4}{(x+3{)}^2}dx=dt{t}^{-4/5+1}\\ & I=\frac{1}{4}\int \frac{1}{t^{4/5}}dt=\frac{1}{4}\frac{t^{1/5}}{1/5}+c\\ & I=\frac{5}{4}{\left(\frac{x-1}{x+3}\right)}^{1/5}+C\\ & A=\frac{5}{4}\alpha =\beta =1B=\frac{1}{5}\\ & \alpha +\beta +20AB=2+20\times \frac{5}{4}\times \frac{1}{5}=7\end{array}$