\[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx \]
Optimal antiderivative \[ -\frac {d^{3} \left (-b e +c d \right )^{3}}{6 e^{7} \left (e x +d \right )^{6}}+\frac {3 d^{2} \left (-b e +c d \right )^{2} \left (-b e +2 c d \right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 d \left (-b e +c d \right ) \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{4 e^{7} \left (e x +d \right )^{4}}+\frac {\left (-b e +2 c d \right ) \left (b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}\right )}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 c \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}+\frac {3 c^{2} \left (-b e +2 c d \right )}{e^{7} \left (e x +d \right )}+\frac {c^{3} \ln \! \left (e x +d \right )}{e^{7}} \]
command
integrate((c*x**2+b*x)**3/(e*x+d)**7,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {c^{3} \log {\left (d + e x \right )}}{e^{7}} + \frac {- b^{3} d^{3} e^{3} - 6 b^{2} c d^{4} e^{2} - 30 b c^{2} d^{5} e + 147 c^{3} d^{6} + x^{5} \left (- 180 b c^{2} e^{6} + 360 c^{3} d e^{5}\right ) + x^{4} \left (- 90 b^{2} c e^{6} - 450 b c^{2} d e^{5} + 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 20 b^{3} e^{6} - 120 b^{2} c d e^{5} - 600 b c^{2} d^{2} e^{4} + 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 15 b^{3} d e^{5} - 90 b^{2} c d^{2} e^{4} - 450 b c^{2} d^{3} e^{3} + 1875 c^{3} d^{4} e^{2}\right ) + x \left (- 6 b^{3} d^{2} e^{4} - 36 b^{2} c d^{3} e^{3} - 180 b c^{2} d^{4} e^{2} + 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \]