\[ \int \frac {(d+e x)^7}{\left (b x+c x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {d^{7}}{2 b^{3} x^{2}}+\frac {d^{6} \left (-7 b e +3 c d \right )}{b^{4} x}+\frac {e^{6} \left (-3 b e +7 c d \right ) x}{c^{4}}+\frac {e^{7} x^{2}}{2 c^{3}}+\frac {\left (-b e +c d \right )^{7}}{2 b^{3} c^{5} \left (c x +b \right )^{2}}+\frac {\left (-b e +c d \right )^{6} \left (4 b e +3 c d \right )}{b^{4} c^{5} \left (c x +b \right )}+\frac {3 d^{5} \left (7 b^{2} e^{2}-7 b c d e +2 c^{2} d^{2}\right ) \ln \! \left (x \right )}{b^{5}}-\frac {3 \left (-b e +c d \right )^{5} \left (2 b^{2} e^{2}+3 b c d e +2 c^{2} d^{2}\right ) \ln \! \left (c x +b \right )}{b^{5} c^{5}} \]
command
integrate((e*x+d)**7/(c*x**2+b*x)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ x \left (- \frac {3 b e^{7}}{c^{4}} + \frac {7 d e^{6}}{c^{3}}\right ) + \frac {- b^{3} c^{5} d^{7} + x^{3} \left (8 b^{7} c e^{7} - 42 b^{6} c^{2} d e^{6} + 84 b^{5} c^{3} d^{2} e^{5} - 70 b^{4} c^{4} d^{3} e^{4} + 42 b^{2} c^{6} d^{5} e^{2} - 42 b c^{7} d^{6} e + 12 c^{8} d^{7}\right ) + x^{2} \left (7 b^{8} e^{7} - 35 b^{7} c d e^{6} + 63 b^{6} c^{2} d^{2} e^{5} - 35 b^{5} c^{3} d^{3} e^{4} - 35 b^{4} c^{4} d^{4} e^{3} + 63 b^{3} c^{5} d^{5} e^{2} - 63 b^{2} c^{6} d^{6} e + 18 b c^{7} d^{7}\right ) + x \left (- 14 b^{3} c^{5} d^{6} e + 4 b^{2} c^{6} d^{7}\right )}{2 b^{6} c^{5} x^{2} + 4 b^{5} c^{6} x^{3} + 2 b^{4} c^{7} x^{4}} + \frac {e^{7} x^{2}}{2 c^{3}} + \frac {3 d^{5} \left (7 b^{2} e^{2} - 7 b c d e + 2 c^{2} d^{2}\right ) \log {\left (x + \frac {- 21 b^{3} c^{4} d^{5} e^{2} + 21 b^{2} c^{5} d^{6} e - 6 b c^{6} d^{7} + 3 b c^{4} d^{5} \left (7 b^{2} e^{2} - 7 b c d e + 2 c^{2} d^{2}\right )}{6 b^{7} e^{7} - 21 b^{6} c d e^{6} + 21 b^{5} c^{2} d^{2} e^{5} - 42 b^{2} c^{5} d^{5} e^{2} + 42 b c^{6} d^{6} e - 12 c^{7} d^{7}} \right )}}{b^{5}} + \frac {3 \left (b e - c d\right )^{5} \left (2 b^{2} e^{2} + 3 b c d e + 2 c^{2} d^{2}\right ) \log {\left (x + \frac {- 21 b^{3} c^{4} d^{5} e^{2} + 21 b^{2} c^{5} d^{6} e - 6 b c^{6} d^{7} + \frac {3 b \left (b e - c d\right )^{5} \left (2 b^{2} e^{2} + 3 b c d e + 2 c^{2} d^{2}\right )}{c}}{6 b^{7} e^{7} - 21 b^{6} c d e^{6} + 21 b^{5} c^{2} d^{2} e^{5} - 42 b^{2} c^{5} d^{5} e^{2} + 42 b c^{6} d^{6} e - 12 c^{7} d^{7}} \right )}}{b^{5} c^{5}} \]