\[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a e +b d \right )^{7}}{5 e^{8} \left (e x +d \right )^{\frac {5}{2}}}-\frac {14 b \left (-a e +b d \right )^{6}}{3 e^{8} \left (e x +d \right )^{\frac {3}{2}}}-\frac {70 b^{4} \left (-a e +b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3 e^{8}}+\frac {42 b^{5} \left (-a e +b d \right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5 e^{8}}-\frac {2 b^{6} \left (-a e +b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{e^{8}}+\frac {2 b^{7} \left (e x +d \right )^{\frac {9}{2}}}{9 e^{8}}+\frac {42 b^{2} \left (-a e +b d \right )^{5}}{e^{8} \sqrt {e x +d}}+\frac {70 b^{3} \left (-a e +b d \right )^{4} \sqrt {e x +d}}{e^{8}} \]
command
integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(7/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {2 b^{7} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{8}} - \frac {42 b^{2} \left (a e - b d\right )^{5}}{e^{8} \sqrt {d + e x}} - \frac {14 b \left (a e - b d\right )^{6}}{3 e^{8} \left (d + e x\right )^{\frac {3}{2}}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (14 a b^{6} e - 14 b^{7} d\right )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (42 a^{2} b^{5} e^{2} - 84 a b^{6} d e + 42 b^{7} d^{2}\right )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (70 a^{3} b^{4} e^{3} - 210 a^{2} b^{5} d e^{2} + 210 a b^{6} d^{2} e - 70 b^{7} d^{3}\right )}{3 e^{8}} + \frac {\sqrt {d + e x} \left (70 a^{4} b^{3} e^{4} - 280 a^{3} b^{4} d e^{3} + 420 a^{2} b^{5} d^{2} e^{2} - 280 a b^{6} d^{3} e + 70 b^{7} d^{4}\right )}{e^{8}} - \frac {2 \left (a e - b d\right )^{7}}{5 e^{8} \left (d + e x\right )^{\frac {5}{2}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________