Calculate Indefinite Integral $\int \frac{x^5(1-x^6)}{(1+x)^{18}}dx$ The following integration is given by Wolfram Alpha
$$\int \frac{x^5(1-x^6)}{(1+x)^{18}}dx=\frac{x^6(28x^4+16x^3+39x^2+16x+28)}{168(1+x)^{16}}.$$
My question is: what is the best (meaning least work), method to achieve this result by hand ? There are two approaches I see, partial fractions, maybe setting $y=1+x$, but this is still alot of work. Or perform successive integration by parts. Both still involve a lot of calculation. The compact form of the answer makes me hope there is a nice way to achieve it. Guessing the ultimate form and then calculating the derivative is, to me, a less desirable method.

$$\int \frac{x^5 \left(1-x^6\right)}{(x+1)^{18}}\,dx=\frac{P(x)}{(x+1)^{17}}$$
$$\frac{x^5 \left(1-x^6\right)}{(x+1)^{18}}=\frac{(x+1) P'(x)-17 P(x)}{(x+1)^{18}}$$ So, $P(x)$ is a polynomial of degree $11$.
Let
$$P(x)=\sum_{n=0}^{11} a_n\,x^n$$ Expand and group powers to get
$$0=(a_1-17 a_0)+(2 a_2-16 a_1) x+(3 a_3-15 a_2) x^2+(4 a_4-14 a_3) x^3+$$ $$(5
   a_5-13 a_4) x^4+(-12 a_5+6 a_6-1) x^5+(7 a_7-11 a_6) x^6+(8 a_8-10 a_7)
   x^7+$$ $$(9 a_9-9 a_8) x^8+(10 a_{10}-8 a_9) x^9+(11 a_{11}-7 a_{10}) x^{10}+(1-6
   a_{11}) x^{11}$$ which is simple.
If I not wrong the coefficients are
$$\left\{0,0,0,0,0,0,\frac{1}{6},\frac{11}{42},\frac{55}{168},\frac{55}{168},\frac{11}{42},\frac{1}{6}\right\}$$ and factoring
$$P(x)=\frac{1}{168} x^6 (x+1) \left(28 x^4+16 x^3+39 x^2+16 x+28\right)$$
