∫ e x ( x + 3 ) ( x + 5 ) 3 d x =

Given that

$\int \frac{e^x(x+3)}{(x+5{)}^3}dx$

$=\int \frac{e^x(x+5-2)}{(x+5{)}^3}dx$

$=\int \frac{e^x\left[(x+5)-2\right]}{(x+5{)}^3}dx$

$=\int e^x \left[ \frac{1}{{\left(x+5\right)}^2 }-\frac{2}{{\left(x+5\right)}^3}\right]dx$

We know that $\int e^x \left(f\left(x\right) +f^{\text{'}}\left(x\right) \right)dx = e^{x}f\left(x\right) + c$

Here $f\left(x\right) = \frac{1}{{\left(x+5\right)}^2} \& f^{\text{'}}\left(x\right) = \frac{-2}{{\left(x+5\right)}^3}$

$=\frac{e^x}{{\left(x+5\right)}^2} + c.$