\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a e +b d \right )^{5} \left (-6 A b e -B a e +7 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e^{8}}-\frac {6 b \left (-a e +b d \right )^{4} \left (-5 A b e -2 B a e +7 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{8}}+\frac {10 b^{2} \left (-a e +b d \right )^{3} \left (-4 A b e -3 B a e +7 B b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{8}}-\frac {10 b^{3} \left (-a e +b d \right )^{2} \left (-3 A b e -4 B a e +7 B b d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9 e^{8}}+\frac {6 b^{4} \left (-a e +b d \right ) \left (-2 A b e -5 B a e +7 B b d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11 e^{8}}-\frac {2 b^{5} \left (-A b e -6 B a e +7 B b d \right ) \left (e x +d \right )^{\frac {13}{2}}}{13 e^{8}}+\frac {2 b^{6} B \left (e x +d \right )^{\frac {15}{2}}}{15 e^{8}}-\frac {2 \left (-a e +b d \right )^{6} \left (-A e +B d \right ) \sqrt {e x +d}}{e^{8}} \]
command
integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________