Computer algebra systems sometimes need a helping hand from human beings. Ask your CAS to evaluate $$\int {2^x \sqrt{4^x-1}}\, dx$$ If it doesn't return an answer, ask it to try $$\int {2^x \sqrt{2^{2x}-1}}\, dx$$ instead. (Side question: If your CAS could not do the first integral, why do you think it was more successful with the second form of the integrand?)

Note: Use an upper-case "C" for the constant of integration.