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.

Explore the embedded symmetry by denoting
$$f_{n,m}(x)= \frac{x^{n-m}+x^{n+m}}{(1+x)^{2n}}$$
which differentiates as
$$f’_{n,m}(x)
=\frac{(n-m)(x^{-m-1}-x^{m+1}) -2m( x^{-m}-x^{m})-(n+m)(x^{-m+1}-x^{m-1}) }{x^{-n}(1+x)^{2n+2}}
$$
In particular, with $n=8$ and $m=0,1,2$
\begin{align}
f’_{8,0}(x)&= \frac{16(x^7-x^9)}{(1+x)^{18}}\\
 f’_{8,1}(x)&= \frac{-2(x^7-x^9) +7(x^6-x^{10})}{(1+x)^{18}}\\
 f’_{8,2}(x)& 
= \frac{-10(x^7-x^9)-4(x^6-x^{10})+ 6(x^5-x^{11})}{(1+x)^{18}}
\end{align}
which leads to
$$
\frac{x^5-x^{11}}{(1+x)^{18}}
= \frac16 f’_{8,2}(x)+\frac2{21} f’_{8,1}(x)+\frac{13}{112} f’_{8,0}(x)
$$
Integrate to obtain
\begin{align}
\int \frac{x^5(1-x^6)}{(1+x)^{18}}dx
&= \frac16 f_{8,2}(x)+\frac2{21} f_{8,1}(x)+\frac{13}{112} f_{8,0}(x)\\
 &= \frac{x^6+x^{10}}{6(1+x)^{16}} + \frac{2(x^7+x^{9})}{21(1+x)^{16}} + \frac{26x^8}{112(1+x)^{16}}\\
 &=\frac{x^6(28x^4+16x^3+39x^2+16x+28)}{168(1+x)^{16}}\\
\end{align}
