\[ \int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx \]
Optimal antiderivative \[ \frac {210 d^{4} \left (-a d +b c \right )^{6} x}{b^{10}}-\frac {\left (-a d +b c \right )^{10}}{3 b^{11} \left (b x +a \right )^{3}}-\frac {5 d \left (-a d +b c \right )^{9}}{b^{11} \left (b x +a \right )^{2}}-\frac {45 d^{2} \left (-a d +b c \right )^{8}}{b^{11} \left (b x +a \right )}+\frac {126 d^{5} \left (-a d +b c \right )^{5} \left (b x +a \right )^{2}}{b^{11}}+\frac {70 d^{6} \left (-a d +b c \right )^{4} \left (b x +a \right )^{3}}{b^{11}}+\frac {30 d^{7} \left (-a d +b c \right )^{3} \left (b x +a \right )^{4}}{b^{11}}+\frac {9 d^{8} \left (-a d +b c \right )^{2} \left (b x +a \right )^{5}}{b^{11}}+\frac {5 d^{9} \left (-a d +b c \right ) \left (b x +a \right )^{6}}{3 b^{11}}+\frac {d^{10} \left (b x +a \right )^{7}}{7 b^{11}}+\frac {120 d^{3} \left (-a d +b c \right )^{7} \ln \! \left (b x +a \right )}{b^{11}} \]
command
integrate((d*x+c)**10/(b*x+a)**4,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ x^{6} \left (- \frac {2 a d^{10}}{3 b^{5}} + \frac {5 c d^{9}}{3 b^{4}}\right ) + x^{5} \left (\frac {2 a^{2} d^{10}}{b^{6}} - \frac {8 a c d^{9}}{b^{5}} + \frac {9 c^{2} d^{8}}{b^{4}}\right ) + x^{4} \left (- \frac {5 a^{3} d^{10}}{b^{7}} + \frac {25 a^{2} c d^{9}}{b^{6}} - \frac {45 a c^{2} d^{8}}{b^{5}} + \frac {30 c^{3} d^{7}}{b^{4}}\right ) + x^{3} \left (\frac {35 a^{4} d^{10}}{3 b^{8}} - \frac {200 a^{3} c d^{9}}{3 b^{7}} + \frac {150 a^{2} c^{2} d^{8}}{b^{6}} - \frac {160 a c^{3} d^{7}}{b^{5}} + \frac {70 c^{4} d^{6}}{b^{4}}\right ) + x^{2} \left (- \frac {28 a^{5} d^{10}}{b^{9}} + \frac {175 a^{4} c d^{9}}{b^{8}} - \frac {450 a^{3} c^{2} d^{8}}{b^{7}} + \frac {600 a^{2} c^{3} d^{7}}{b^{6}} - \frac {420 a c^{4} d^{6}}{b^{5}} + \frac {126 c^{5} d^{5}}{b^{4}}\right ) + x \left (\frac {84 a^{6} d^{10}}{b^{10}} - \frac {560 a^{5} c d^{9}}{b^{9}} + \frac {1575 a^{4} c^{2} d^{8}}{b^{8}} - \frac {2400 a^{3} c^{3} d^{7}}{b^{7}} + \frac {2100 a^{2} c^{4} d^{6}}{b^{6}} - \frac {1008 a c^{5} d^{5}}{b^{5}} + \frac {210 c^{6} d^{4}}{b^{4}}\right ) + \frac {- 121 a^{10} d^{10} + 955 a^{9} b c d^{9} - 3285 a^{8} b^{2} c^{2} d^{8} + 6420 a^{7} b^{3} c^{3} d^{7} - 7770 a^{6} b^{4} c^{4} d^{6} + 5922 a^{5} b^{5} c^{5} d^{5} - 2730 a^{4} b^{6} c^{6} d^{4} + 660 a^{3} b^{7} c^{7} d^{3} - 45 a^{2} b^{8} c^{8} d^{2} - 5 a b^{9} c^{9} d - b^{10} c^{10} + x^{2} \left (- 135 a^{8} b^{2} d^{10} + 1080 a^{7} b^{3} c d^{9} - 3780 a^{6} b^{4} c^{2} d^{8} + 7560 a^{5} b^{5} c^{3} d^{7} - 9450 a^{4} b^{6} c^{4} d^{6} + 7560 a^{3} b^{7} c^{5} d^{5} - 3780 a^{2} b^{8} c^{6} d^{4} + 1080 a b^{9} c^{7} d^{3} - 135 b^{10} c^{8} d^{2}\right ) + x \left (- 255 a^{9} b d^{10} + 2025 a^{8} b^{2} c d^{9} - 7020 a^{7} b^{3} c^{2} d^{8} + 13860 a^{6} b^{4} c^{3} d^{7} - 17010 a^{5} b^{5} c^{4} d^{6} + 13230 a^{4} b^{6} c^{5} d^{5} - 6300 a^{3} b^{7} c^{6} d^{4} + 1620 a^{2} b^{8} c^{7} d^{3} - 135 a b^{9} c^{8} d^{2} - 15 b^{10} c^{9} d\right )}{3 a^{3} b^{11} + 9 a^{2} b^{12} x + 9 a b^{13} x^{2} + 3 b^{14} x^{3}} + \frac {d^{10} x^{7}}{7 b^{4}} - \frac {120 d^{3} \left (a d - b c\right )^{7} \log {\left (a + b x \right )}}{b^{11}} \]