The rate at which a submersible drone consumes battery power, $P(t)$ in watts, is modeled by the function:

$$P(t)=20t^3-\frac{6}{5t^4}+\frac{9}{2}$$

where $t>0$ is the duration of the mission in hours. Find an expression for the total energy consumption by evaluating the indefinite integral:

$$\int \left(20t^3-\frac{6}{5t^4}+\frac{9}{2}\right)dt$$

giving each term in its simplest form.