some Indefinite Integrals.

(1)::\int \frac{\left ( \frac{25}{9} \right )^{x}}{\left ( \frac{15}{9} \right )^{x}-1} dx

\int \frac{\left ( \frac{5}{3} \right )^{2x}}{\left ( \frac{5}{3} \right )^{x}-1} dx

let\, \, \left ( \frac{5}{3} \right )^{x} =t \Rightarrow \left ( \frac{5}{3} \right )^{x} log\left ( \frac{5}{3} \right )dx =dt

integral\, \, is \, \, \frac{ 1}{log\left ( \frac{5}{3} \right )}\int \frac{t^{2}}{(t-1)t}dt

\frac{1}{log\left ( \frac{5}{3} \right )}\int \frac{tdt}{t-1}

which can be easily done

For second, Partial Fractions method should work efficiently.

PS Note that I have posted answers from Wolfram only after solving it. It gives it in a neat form.