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?

The following is a close relative of integration by parts. Let us guess that the answer is $x^5e^x$. Differentiate to see whether we got lucky. We get $x^5e^x+5x^4e^x$. Too bad!
But maybe we can fix things by subtracting $5x^4e^x$, so our next guess is $x^5e^x -5x^4e^x$. Differentiate. We get $x^5e^x -20x^3e^x$. Maybe we can fix things by  adding
$20x^3e^x$. So our next guess is $x^5e^x-5x^4e^x +20x^3e^x$. Continue. It is soon over.  
Remark: The general idea works nicely for $\sin(5x)e^{-3x}$, a standard integration by parts problem that causes me a great deal of trouble because of all the minus signs. But if we make a suitable guess, we can quickly arrive at the answer.
