What is the most efficient method to evaluate this indefinite integral? $$\int x^5 e^x\,\mathrm{d}x$$
Is there another, more efficient way to solve this integral that is not integration by parts?

You're question asked if there is a more efficient way than integration by parts to solve the indefinite integral $\int x^5e^x dx$, and other users have provided good answers.
But in case you tacitly assumed that the answer to your question would pretty much be the same for definite integrals like $\int_a^b x^n e^{kx} dx$, I'd like to show for contrast that in this case there can be much more efficient methods than IBP for finding the integral.
For example, consider the integral $\int_0^\infty x^5 e^{-x}dx$. To find this integral, first evaluate $\int_0^\infty e^{-ax}dx$:
$$\int_0^\infty e^{-ax}dx=-\frac{e^{-ax}}{a}\bigg{|}_0^\infty=\frac{1}{a}.$$
Then just differentiate both sides with respect to $a$ five times to get:
$$-\int_0^\infty x^5 e^{-ax}dx=-\frac{120}{a^6}.$$
Finally, set $a=1$ and voila, the integral is $\int_0^\infty x^5 e^{-x}dx=120$. And all you had to do really was a single integration plus five differentiations.
