\[ \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-a d +b c \right )^{3} \left (-c f +d e \right ) \left (f x +e \right )^{\frac {3}{2}}}{3 d^{5}}-\frac {2 \left (-a d +b c \right )^{3} \left (f x +e \right )^{\frac {5}{2}}}{5 d^{4}}+\frac {2 b \left (3 a^{2} d^{2} f^{2}-3 a b d f \left (c f +d e \right )+b^{2} \left (c^{2} f^{2}+c d e f +d^{2} e^{2}\right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7 d^{3} f^{3}}-\frac {2 b^{2} \left (-3 a d f +b c f +2 b d e \right ) \left (f x +e \right )^{\frac {9}{2}}}{9 d^{2} f^{3}}+\frac {2 b^{3} \left (f x +e \right )^{\frac {11}{2}}}{11 d \,f^{3}}+\frac {2 \left (-a d +b c \right )^{3} \left (-c f +d e \right )^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {d}\, \sqrt {f x +e}}{\sqrt {-c f +d e}}\right )}{d^{\frac {13}{2}}}-\frac {2 \left (-a d +b c \right )^{3} \left (-c f +d e \right )^{2} \sqrt {f x +e}}{d^{6}} \]
command
integrate((b*x+a)**3*(f*x+e)**(5/2)/(d*x+c),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {2 b^{3} \left (e + f x\right )^{\frac {11}{2}}}{11 d f^{3}} + \frac {\left (e + f x\right )^{\frac {9}{2}} \cdot \left (6 a b^{2} d f - 2 b^{3} c f - 4 b^{3} d e\right )}{9 d^{2} f^{3}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (6 a^{2} b d^{2} f^{2} - 6 a b^{2} c d f^{2} - 6 a b^{2} d^{2} e f + 2 b^{3} c^{2} f^{2} + 2 b^{3} c d e f + 2 b^{3} d^{2} e^{2}\right )}{7 d^{3} f^{3}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \cdot \left (2 a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}\right )}{5 d^{4}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (- 2 a^{3} c d^{3} f + 2 a^{3} d^{4} e + 6 a^{2} b c^{2} d^{2} f - 6 a^{2} b c d^{3} e - 6 a b^{2} c^{3} d f + 6 a b^{2} c^{2} d^{2} e + 2 b^{3} c^{4} f - 2 b^{3} c^{3} d e\right )}{3 d^{5}} + \frac {\sqrt {e + f x} \left (2 a^{3} c^{2} d^{3} f^{2} - 4 a^{3} c d^{4} e f + 2 a^{3} d^{5} e^{2} - 6 a^{2} b c^{3} d^{2} f^{2} + 12 a^{2} b c^{2} d^{3} e f - 6 a^{2} b c d^{4} e^{2} + 6 a b^{2} c^{4} d f^{2} - 12 a b^{2} c^{3} d^{2} e f + 6 a b^{2} c^{2} d^{3} e^{2} - 2 b^{3} c^{5} f^{2} + 4 b^{3} c^{4} d e f - 2 b^{3} c^{3} d^{2} e^{2}\right )}{d^{6}} - \frac {2 \left (a d - b c\right )^{3} \left (c f - d e\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{7} \sqrt {\frac {c f - d e}{d}}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________