\[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-a d +b c \right ) \left (a^{2} d^{2} \left (C d -3 c D\right )-a b d \left (-3 B \,d^{2}+8 c C d -15 c^{2} D\right )+b^{2} \left (3 A \,d^{3}-6 B c \,d^{2}+10 c^{2} C d -15 c^{3} D\right )\right ) \left (d x +c \right )^{\frac {3}{2}}}{3 d^{7}}+\frac {2 \left (a^{3} d^{3} D+3 a^{2} b \,d^{2} \left (C d -4 c D\right )-3 a \,b^{2} d \left (-B \,d^{2}+4 c C d -10 c^{2} D\right )+b^{3} \left (A \,d^{3}-4 B c \,d^{2}+10 c^{2} C d -20 c^{3} D\right )\right ) \left (d x +c \right )^{\frac {5}{2}}}{5 d^{7}}+\frac {2 b \left (3 a^{2} d^{2} D+3 a b d \left (C d -5 c D\right )-b^{2} \left (-B \,d^{2}+5 c C d -15 c^{2} D\right )\right ) \left (d x +c \right )^{\frac {7}{2}}}{7 d^{7}}+\frac {2 b^{2} \left (b C d +3 a d D-6 b c D\right ) \left (d x +c \right )^{\frac {9}{2}}}{9 d^{7}}+\frac {2 b^{3} D \left (d x +c \right )^{\frac {11}{2}}}{11 d^{7}}+\frac {2 \left (-a d +b c \right )^{3} \left (A \,d^{3}-B c \,d^{2}+c^{2} C d -c^{3} D\right )}{d^{7} \sqrt {d x +c}}-\frac {2 \left (-a d +b c \right )^{2} \left (a d \left (-B \,d^{2}+2 c C d -3 c^{2} D\right )-b \left (3 A \,d^{3}-4 B c \,d^{2}+5 c^{2} C d -6 c^{3} D\right )\right ) \sqrt {d x +c}}{d^{7}} \]
command
integrate((b*x+a)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {2 D b^{3} \left (c + d x\right )^{\frac {11}{2}}}{11 d^{7}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (2 C b^{3} d + 6 D a b^{2} d - 12 D b^{3} c\right )}{9 d^{7}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (2 B b^{3} d^{2} + 6 C a b^{2} d^{2} - 10 C b^{3} c d + 6 D a^{2} b d^{2} - 30 D a b^{2} c d + 30 D b^{3} c^{2}\right )}{7 d^{7}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (2 A b^{3} d^{3} + 6 B a b^{2} d^{3} - 8 B b^{3} c d^{2} + 6 C a^{2} b d^{3} - 24 C a b^{2} c d^{2} + 20 C b^{3} c^{2} d + 2 D a^{3} d^{3} - 24 D a^{2} b c d^{2} + 60 D a b^{2} c^{2} d - 40 D b^{3} c^{3}\right )}{5 d^{7}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (6 A a b^{2} d^{4} - 6 A b^{3} c d^{3} + 6 B a^{2} b d^{4} - 18 B a b^{2} c d^{3} + 12 B b^{3} c^{2} d^{2} + 2 C a^{3} d^{4} - 18 C a^{2} b c d^{3} + 36 C a b^{2} c^{2} d^{2} - 20 C b^{3} c^{3} d - 6 D a^{3} c d^{3} + 36 D a^{2} b c^{2} d^{2} - 60 D a b^{2} c^{3} d + 30 D b^{3} c^{4}\right )}{3 d^{7}} + \frac {\sqrt {c + d x} \left (6 A a^{2} b d^{5} - 12 A a b^{2} c d^{4} + 6 A b^{3} c^{2} d^{3} + 2 B a^{3} d^{5} - 12 B a^{2} b c d^{4} + 18 B a b^{2} c^{2} d^{3} - 8 B b^{3} c^{3} d^{2} - 4 C a^{3} c d^{4} + 18 C a^{2} b c^{2} d^{3} - 24 C a b^{2} c^{3} d^{2} + 10 C b^{3} c^{4} d + 6 D a^{3} c^{2} d^{3} - 24 D a^{2} b c^{3} d^{2} + 30 D a b^{2} c^{4} d - 12 D b^{3} c^{5}\right )}{d^{7}} + \frac {2 \left (a d - b c\right )^{3} \left (- A d^{3} + B c d^{2} - C c^{2} d + D c^{3}\right )}{d^{7} \sqrt {c + d x}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________