\[ \int \frac {(d+e x)^7}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {e^{6} \left (-6 a e +7 b d \right ) x}{b^{7}}+\frac {e^{7} x^{2}}{2 b^{6}}-\frac {\left (-a e +b d \right )^{7}}{5 b^{8} \left (b x +a \right )^{5}}-\frac {7 e \left (-a e +b d \right )^{6}}{4 b^{8} \left (b x +a \right )^{4}}-\frac {7 e^{2} \left (-a e +b d \right )^{5}}{b^{8} \left (b x +a \right )^{3}}-\frac {35 e^{3} \left (-a e +b d \right )^{4}}{2 b^{8} \left (b x +a \right )^{2}}-\frac {35 e^{4} \left (-a e +b d \right )^{3}}{b^{8} \left (b x +a \right )}+\frac {21 e^{5} \left (-a e +b d \right )^{2} \ln \! \left (b x +a \right )}{b^{8}} \]
command
integrate((e*x+d)**7/(b**2*x**2+2*a*b*x+a**2)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ x \left (- \frac {6 a e^{7}}{b^{7}} + \frac {7 d e^{6}}{b^{6}}\right ) + \frac {459 a^{7} e^{7} - 1218 a^{6} b d e^{6} + 959 a^{5} b^{2} d^{2} e^{5} - 140 a^{4} b^{3} d^{3} e^{4} - 35 a^{3} b^{4} d^{4} e^{3} - 14 a^{2} b^{5} d^{5} e^{2} - 7 a b^{6} d^{6} e - 4 b^{7} d^{7} + x^{4} \left (700 a^{3} b^{4} e^{7} - 2100 a^{2} b^{5} d e^{6} + 2100 a b^{6} d^{2} e^{5} - 700 b^{7} d^{3} e^{4}\right ) + x^{3} \left (2450 a^{4} b^{3} e^{7} - 7000 a^{3} b^{4} d e^{6} + 6300 a^{2} b^{5} d^{2} e^{5} - 1400 a b^{6} d^{3} e^{4} - 350 b^{7} d^{4} e^{3}\right ) + x^{2} \left (3290 a^{5} b^{2} e^{7} - 9100 a^{4} b^{3} d e^{6} + 7700 a^{3} b^{4} d^{2} e^{5} - 1400 a^{2} b^{5} d^{3} e^{4} - 350 a b^{6} d^{4} e^{3} - 140 b^{7} d^{5} e^{2}\right ) + x \left (1995 a^{6} b e^{7} - 5390 a^{5} b^{2} d e^{6} + 4375 a^{4} b^{3} d^{2} e^{5} - 700 a^{3} b^{4} d^{3} e^{4} - 175 a^{2} b^{5} d^{4} e^{3} - 70 a b^{6} d^{5} e^{2} - 35 b^{7} d^{6} e\right )}{20 a^{5} b^{8} + 100 a^{4} b^{9} x + 200 a^{3} b^{10} x^{2} + 200 a^{2} b^{11} x^{3} + 100 a b^{12} x^{4} + 20 b^{13} x^{5}} + \frac {e^{7} x^{2}}{2 b^{6}} + \frac {21 e^{5} \left (a e - b d\right )^{2} \log {\left (a + b x \right )}}{b^{8}} \]