17.13 Problem number 1527

\[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \]

Optimal antiderivative \[ \frac {e^{6} x}{b^{6}}-\frac {\left (-a e +b d \right )^{6}}{5 b^{7} \left (b x +a \right )^{5}}-\frac {3 e \left (-a e +b d \right )^{5}}{2 b^{7} \left (b x +a \right )^{4}}-\frac {5 e^{2} \left (-a e +b d \right )^{4}}{b^{7} \left (b x +a \right )^{3}}-\frac {10 e^{3} \left (-a e +b d \right )^{3}}{b^{7} \left (b x +a \right )^{2}}-\frac {15 e^{4} \left (-a e +b d \right )^{2}}{b^{7} \left (b x +a \right )}+\frac {6 e^{5} \left (-a e +b d \right ) \ln \! \left (b x +a \right )}{b^{7}} \]

command

integrate((e*x+d)**6/(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

\[ \frac {- 87 a^{6} e^{6} + 137 a^{5} b d e^{5} - 30 a^{4} b^{2} d^{2} e^{4} - 10 a^{3} b^{3} d^{3} e^{3} - 5 a^{2} b^{4} d^{4} e^{2} - 3 a b^{5} d^{5} e - 2 b^{6} d^{6} + x^{4} \left (- 150 a^{2} b^{4} e^{6} + 300 a b^{5} d e^{5} - 150 b^{6} d^{2} e^{4}\right ) + x^{3} \left (- 500 a^{3} b^{3} e^{6} + 900 a^{2} b^{4} d e^{5} - 300 a b^{5} d^{2} e^{4} - 100 b^{6} d^{3} e^{3}\right ) + x^{2} \left (- 650 a^{4} b^{2} e^{6} + 1100 a^{3} b^{3} d e^{5} - 300 a^{2} b^{4} d^{2} e^{4} - 100 a b^{5} d^{3} e^{3} - 50 b^{6} d^{4} e^{2}\right ) + x \left (- 385 a^{5} b e^{6} + 625 a^{4} b^{2} d e^{5} - 150 a^{3} b^{3} d^{2} e^{4} - 50 a^{2} b^{4} d^{3} e^{3} - 25 a b^{5} d^{4} e^{2} - 15 b^{6} d^{5} e\right )}{10 a^{5} b^{7} + 50 a^{4} b^{8} x + 100 a^{3} b^{9} x^{2} + 100 a^{2} b^{10} x^{3} + 50 a b^{11} x^{4} + 10 b^{12} x^{5}} + \frac {e^{6} x}{b^{6}} - \frac {6 e^{5} \left (a e - b d\right ) \log {\left (a + b x \right )}}{b^{7}} \]