\[ \int \frac {A+B x}{(d+e x) \left (a+b x+c x^2\right )^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {\frac {2 a B \left (-b e +2 c d \right )}{5}-\frac {2 A \left (2 a c e -b^{2} e +b c d \right )}{5}+\frac {2 c \left (A b e -2 A c d -2 a B e +B b d \right ) x}{5}}{\left (-4 a c +b^{2}\right ) \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}+\frac {\frac {16 a c e \left (-b e +2 c d \right ) \left (A b e -2 A c d -2 a B e +B b d \right )}{15}+\frac {2 \left (2 a c e -b^{2} e +b c d \right ) \left (5 b^{2} e \left (-A e +B d \right )-8 b c d \left (A e +B d \right )+4 c \left (5 A a \,e^{2}+4 A c \,d^{2}-a B d e \right )\right )}{15}+\frac {2 c \left (8 c e \left (-2 a e +b d \right ) \left (A b e -2 A c d -2 a B e +B b d \right )+\left (-b e +2 c d \right ) \left (5 b^{2} e \left (-A e +B d \right )-8 b c d \left (A e +B d \right )+4 c \left (5 A a \,e^{2}+4 A c \,d^{2}-a B d e \right )\right )\right ) x}{15}}{\left (-4 a c +b^{2}\right )^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {e^{5} \left (-A e +B d \right ) \arctanh \left (\frac {b d -2 a e +\left (-b e +2 c d \right ) x}{2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \sqrt {c \,x^{2}+b x +a}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{\frac {7}{2}}}+\frac {\frac {8 a c e \left (-b e +2 c d \right ) \left (8 c e \left (-2 a e +b d \right ) \left (A b e -2 A c d -2 a B e +B b d \right )+\left (-b e +2 c d \right ) \left (5 b^{2} e \left (-A e +B d \right )-8 b c d \left (A e +B d \right )+4 c \left (5 A a \,e^{2}+4 A c \,d^{2}-a B d e \right )\right )\right )}{15}-\frac {2 \left (2 a c e -b^{2} e +b c d \right ) \left (8 c d e \left (A b e -2 A c d -2 a B e +B b d \right ) \left (4 a c e -3 b^{2} e +4 b c d \right )+\left (8 c^{2} d^{2}-3 b^{2} e^{2}-4 c e \left (-3 a e +b d \right )\right ) \left (5 b^{2} e \left (-A e +B d \right )-8 b c d \left (A e +B d \right )+4 c \left (5 A a \,e^{2}+4 A c \,d^{2}-a B d e \right )\right )\right )}{15}+\frac {2 c \left (4 c e \left (-2 a e +b d \right ) \left (8 c e \left (-2 a e +b d \right ) \left (A b e -2 A c d -2 a B e +B b d \right )+\left (-b e +2 c d \right ) \left (5 b^{2} e \left (-A e +B d \right )-8 b c d \left (A e +B d \right )+4 c \left (5 A a \,e^{2}+4 A c \,d^{2}-a B d e \right )\right )\right )-\left (-b e +2 c d \right ) \left (8 c d e \left (A b e -2 A c d -2 a B e +B b d \right ) \left (4 a c e -3 b^{2} e +4 b c d \right )+\left (8 c^{2} d^{2}-3 b^{2} e^{2}-4 c e \left (-3 a e +b d \right )\right ) \left (5 b^{2} e \left (-A e +B d \right )-8 b c d \left (A e +B d \right )+4 c \left (5 A a \,e^{2}+4 A c \,d^{2}-a B d e \right )\right )\right )\right ) x}{15}}{\left (-4 a c +b^{2}\right )^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right )^{3} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((B*x+A)/(e*x+d)/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________