\[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{12}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a d +b c \right )^{3}}{8 b^{4} \left (b x +a \right )^{8}}-\frac {3 d \left (-a d +b c \right )^{2}}{7 b^{4} \left (b x +a \right )^{7}}-\frac {d^{2} \left (-a d +b c \right )}{2 b^{4} \left (b x +a \right )^{6}}-\frac {d^{3}}{5 b^{4} \left (b x +a \right )^{5}} \]
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
\[ \frac {- a^{3} d^{3} - 5 a^{2} b c d^{2} - 15 a b^{2} c^{2} d - 35 b^{3} c^{3} - 56 b^{3} d^{3} x^{3} + x^{2} \left (- 28 a b^{2} d^{3} - 140 b^{3} c d^{2}\right ) + x \left (- 8 a^{2} b d^{3} - 40 a b^{2} c d^{2} - 120 b^{3} c^{2} d\right )}{280 a^{8} b^{4} + 2240 a^{7} b^{5} x + 7840 a^{6} b^{6} x^{2} + 15680 a^{5} b^{7} x^{3} + 19600 a^{4} b^{8} x^{4} + 15680 a^{3} b^{9} x^{5} + 7840 a^{2} b^{10} x^{6} + 2240 a b^{11} x^{7} + 280 b^{12} x^{8}} \]