\[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-a d +b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+c^{2} C d -c^{3} D\right )}{3 d^{6} \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 \left (a^{2} d^{2} D+2 a b d \left (C d -4 c D\right )-b^{2} \left (-B \,d^{2}+4 c C d -10 c^{2} D\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3 d^{6}}+\frac {2 b \left (b C d +2 a d D-5 b c D\right ) \left (d x +c \right )^{\frac {5}{2}}}{5 d^{6}}+\frac {2 b^{2} D \left (d x +c \right )^{\frac {7}{2}}}{7 d^{6}}-\frac {2 \left (-a d +b c \right ) \left (a d \left (-B \,d^{2}+2 c C d -3 c^{2} D\right )-b \left (2 A \,d^{3}-3 B c \,d^{2}+4 c^{2} C d -5 c^{3} D\right )\right )}{d^{6} \sqrt {d x +c}}+\frac {2 \left (a^{2} d^{2} \left (C d -3 c D\right )-2 a b d \left (-B \,d^{2}+3 c C d -6 c^{2} D\right )+b^{2} \left (A \,d^{3}-3 B c \,d^{2}+6 c^{2} C d -10 c^{3} D\right )\right ) \sqrt {d x +c}}{d^{6}} \]
command
integrate((b*x+a)**2*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {2 D b^{2} \left (c + d x\right )^{\frac {7}{2}}}{7 d^{6}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (2 C b^{2} d + 4 D a b d - 10 D b^{2} c\right )}{5 d^{6}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (2 B b^{2} d^{2} + 4 C a b d^{2} - 8 C b^{2} c d + 2 D a^{2} d^{2} - 16 D a b c d + 20 D b^{2} c^{2}\right )}{3 d^{6}} + \frac {\sqrt {c + d x} \left (2 A b^{2} d^{3} + 4 B a b d^{3} - 6 B b^{2} c d^{2} + 2 C a^{2} d^{3} - 12 C a b c d^{2} + 12 C b^{2} c^{2} d - 6 D a^{2} c d^{2} + 24 D a b c^{2} d - 20 D b^{2} c^{3}\right )}{d^{6}} - \frac {2 \left (a d - b c\right ) \left (2 A b d^{3} + B a d^{3} - 3 B b c d^{2} - 2 C a c d^{2} + 4 C b c^{2} d + 3 D a c^{2} d - 5 D b c^{3}\right )}{d^{6} \sqrt {c + d x}} + \frac {2 \left (a d - b c\right )^{2} \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{3 d^{6} \left (c + d x\right )^{\frac {3}{2}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________