17.14 Problem number 1798

$$\int \frac{{(ac+(bc+ad)x+bd{x}^2)}^3}{(a+bx{)}^{12}}dx$$

Optimal antiderivative $$-\frac{{(-ad+bc)}^3}{8b^4{(bx+a)}^8}-\frac{3d{(-ad+bc)}^2}{7b^4{(bx+a)}^7}-\frac{d^2(-ad+bc)}{2b^4{(bx+a)}^6}-\frac{d^3}{5b^4{(bx+a)}^5}$$

command

integrate((a*c+(a*d+b*c)*x+b*d*x**2)**3/(b*x+a)**12,x)

Sympy 1.10.1 under Python 3.10.4 output

$$\text{Timed out}$$

Sympy 1.8 under Python 3.8.8 output

$$\frac{-a^3{d}^3-5a^{2}bc{d}^2-15a{b}^2{c}^{2}d-35b^3{c}^3-56b^3{d}^3{x}^3+x^2(-28a{b}^2{d}^3-140b^{3}c{d}^2)+x(-8a^{2}b{d}^3-40a{b}^{2}c{d}^2-120b^3{c}^{2}d)}{280a^8{b}^4+2240a^7{b}^{5}x+7840a^6{b}^6{x}^2+15680a^5{b}^7{x}^3+19600a^4{b}^8{x}^4+15680a^3{b}^9{x}^5+7840a^2{b}^{10}{x}^6+2240a{b}^{11}{x}^7+280b^{12}{x}^8}$$