\[ \int \frac {(a+b x)^5}{(c+d x)^8} \, dx \]
Optimal antiderivative \[ \frac {\left (b x +a \right )^{6}}{7 \left (-a d +b c \right ) \left (d x +c \right )^{7}}+\frac {b \left (b x +a \right )^{6}}{42 \left (-a d +b c \right )^{2} \left (d x +c \right )^{6}} \]
command
integrate((b*x+a)**5/(d*x+c)**8,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {- 6 a^{5} d^{5} - 5 a^{4} b c d^{4} - 4 a^{3} b^{2} c^{2} d^{3} - 3 a^{2} b^{3} c^{3} d^{2} - 2 a b^{4} c^{4} d - b^{5} c^{5} - 21 b^{5} d^{5} x^{5} + x^{4} \left (- 70 a b^{4} d^{5} - 35 b^{5} c d^{4}\right ) + x^{3} \left (- 105 a^{2} b^{3} d^{5} - 70 a b^{4} c d^{4} - 35 b^{5} c^{2} d^{3}\right ) + x^{2} \left (- 84 a^{3} b^{2} d^{5} - 63 a^{2} b^{3} c d^{4} - 42 a b^{4} c^{2} d^{3} - 21 b^{5} c^{3} d^{2}\right ) + x \left (- 35 a^{4} b d^{5} - 28 a^{3} b^{2} c d^{4} - 21 a^{2} b^{3} c^{2} d^{3} - 14 a b^{4} c^{3} d^{2} - 7 b^{5} c^{4} d\right )}{42 c^{7} d^{6} + 294 c^{6} d^{7} x + 882 c^{5} d^{8} x^{2} + 1470 c^{4} d^{9} x^{3} + 1470 c^{3} d^{10} x^{4} + 882 c^{2} d^{11} x^{5} + 294 c d^{12} x^{6} + 42 d^{13} x^{7}} \]