\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 b \left (-a e +b d \right )^{4} \left (-5 A b e -2 B a e +7 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{e^{8}}+\frac {2 b^{2} \left (-a e +b d \right )^{3} \left (-4 A b e -3 B a e +7 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{e^{8}}-\frac {10 b^{3} \left (-a e +b d \right )^{2} \left (-3 A b e -4 B a e +7 B b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{8}}+\frac {2 b^{4} \left (-a e +b d \right ) \left (-2 A b e -5 B a e +7 B b d \right ) \left (e x +d \right )^{\frac {9}{2}}}{3 e^{8}}-\frac {2 b^{5} \left (-A b e -6 B a e +7 B b d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11 e^{8}}+\frac {2 b^{6} B \left (e x +d \right )^{\frac {13}{2}}}{13 e^{8}}+\frac {2 \left (-a e +b d \right )^{6} \left (-A e +B d \right )}{e^{8} \sqrt {e x +d}}+\frac {2 \left (-a e +b d \right )^{5} \left (-6 A b e -B a e +7 B b d \right ) \sqrt {e x +d}}{e^{8}} \]
command
integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {2 B b^{6} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{8}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (2 A b^{6} e + 12 B a b^{5} e - 14 B b^{6} d\right )}{11 e^{8}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (12 A a b^{5} e^{2} - 12 A b^{6} d e + 30 B a^{2} b^{4} e^{2} - 72 B a b^{5} d e + 42 B b^{6} d^{2}\right )}{9 e^{8}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (30 A a^{2} b^{4} e^{3} - 60 A a b^{5} d e^{2} + 30 A b^{6} d^{2} e + 40 B a^{3} b^{3} e^{3} - 150 B a^{2} b^{4} d e^{2} + 180 B a b^{5} d^{2} e - 70 B b^{6} d^{3}\right )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (40 A a^{3} b^{3} e^{4} - 120 A a^{2} b^{4} d e^{3} + 120 A a b^{5} d^{2} e^{2} - 40 A b^{6} d^{3} e + 30 B a^{4} b^{2} e^{4} - 160 B a^{3} b^{3} d e^{3} + 300 B a^{2} b^{4} d^{2} e^{2} - 240 B a b^{5} d^{3} e + 70 B b^{6} d^{4}\right )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (30 A a^{4} b^{2} e^{5} - 120 A a^{3} b^{3} d e^{4} + 180 A a^{2} b^{4} d^{2} e^{3} - 120 A a b^{5} d^{3} e^{2} + 30 A b^{6} d^{4} e + 12 B a^{5} b e^{5} - 90 B a^{4} b^{2} d e^{4} + 240 B a^{3} b^{3} d^{2} e^{3} - 300 B a^{2} b^{4} d^{3} e^{2} + 180 B a b^{5} d^{4} e - 42 B b^{6} d^{5}\right )}{3 e^{8}} + \frac {\sqrt {d + e x} \left (12 A a^{5} b e^{6} - 60 A a^{4} b^{2} d e^{5} + 120 A a^{3} b^{3} d^{2} e^{4} - 120 A a^{2} b^{4} d^{3} e^{3} + 60 A a b^{5} d^{4} e^{2} - 12 A b^{6} d^{5} e + 2 B a^{6} e^{6} - 24 B a^{5} b d e^{5} + 90 B a^{4} b^{2} d^{2} e^{4} - 160 B a^{3} b^{3} d^{3} e^{3} + 150 B a^{2} b^{4} d^{4} e^{2} - 72 B a b^{5} d^{5} e + 14 B b^{6} d^{6}\right )}{e^{8}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )^{6}}{e^{8} \sqrt {d + e x}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________