\[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\left (e x +d \right )^{\frac {5}{2}} \left (a \left (A e +B d \right )+\left (A c d +a B e \right ) x \right )}{4 a c \left (-c \,x^{2}+a \right )^{2}}-\frac {\arctanh \left (\frac {c^{\frac {1}{4}} \sqrt {e x +d}}{\sqrt {e \sqrt {a}+d \sqrt {c}}}\right ) \left (e \sqrt {a}+d \sqrt {c}\right )^{\frac {3}{2}} \left (7 a B e \left (-3 e \sqrt {a}+2 d \sqrt {c}\right )-A \left (12 c^{\frac {3}{2}} d^{2}-18 c d e \sqrt {a}+5 a \,e^{2} \sqrt {c}\right )\right )}{32 a^{\frac {5}{2}} c^{\frac {11}{4}}}+\frac {\arctanh \left (\frac {c^{\frac {1}{4}} \sqrt {e x +d}}{\sqrt {-e \sqrt {a}+d \sqrt {c}}}\right ) \left (-e \sqrt {a}+d \sqrt {c}\right )^{\frac {3}{2}} \left (7 a B e \left (3 e \sqrt {a}+2 d \sqrt {c}\right )-A \left (12 c^{\frac {3}{2}} d^{2}+18 c d e \sqrt {a}+5 a \,e^{2} \sqrt {c}\right )\right )}{32 a^{\frac {5}{2}} c^{\frac {11}{4}}}+\frac {\left (a e \left (-5 A a \,e^{2}+7 A c \,d^{2}-14 a B d e \right )+\left (2 A c d \left (-2 a \,e^{2}+3 c \,d^{2}\right )-7 a B e \left (a \,e^{2}+c \,d^{2}\right )\right ) x \right ) \sqrt {e x +d}}{16 a^{2} c^{2} \left (-c \,x^{2}+a \right )} \]
command
integrate((B*x+A)*(e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} A c^{3} d^{3} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{3} d^{4} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{3} d^{5} e - 6 \, \sqrt {x e + d} A c^{3} d^{6} e - 7 \, {\left (x e + d\right )}^{\frac {7}{2}} B a c^{2} d^{2} e^{2} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B a c^{2} d^{3} e^{2} - 21 \, {\left (x e + d\right )}^{\frac {3}{2}} B a c^{2} d^{4} e^{2} + 7 \, \sqrt {x e + d} B a c^{2} d^{5} e^{2} - 8 \, {\left (x e + d\right )}^{\frac {7}{2}} A a c^{2} d e^{3} + 23 \, {\left (x e + d\right )}^{\frac {5}{2}} A a c^{2} d^{2} e^{3} - 32 \, {\left (x e + d\right )}^{\frac {3}{2}} A a c^{2} d^{3} e^{3} + 17 \, \sqrt {x e + d} A a c^{2} d^{4} e^{3} - 11 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} c e^{4} + 7 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} c d e^{4} + 14 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} c d^{2} e^{4} - 14 \, \sqrt {x e + d} B a^{2} c d^{3} e^{4} - 9 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} c e^{5} + 14 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} c d e^{5} - 16 \, \sqrt {x e + d} A a^{2} c d^{2} e^{5} + 7 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} e^{6} + 7 \, \sqrt {x e + d} B a^{3} d e^{6} + 5 \, \sqrt {x e + d} A a^{3} e^{7}}{16 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )}^{2} a^{2} c^{2}} - \frac {{\left ({\left (6 \, a c^{2} d^{2} e - 12 \, \sqrt {a c} c^{2} d^{3} + 13 \, \sqrt {a c} a c d e^{2} - 5 \, a^{2} c e^{3}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} A {\left | c \right |} + 7 \, {\left (2 \, \sqrt {a c} a c d^{2} e - a^{2} c d e^{2} - 3 \, \sqrt {a c} a^{2} e^{3}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d + \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, a^{3} c^{5}} - \frac {{\left ({\left (6 \, a c^{2} d^{2} e + 12 \, \sqrt {a c} c^{2} d^{3} - 13 \, \sqrt {a c} a c d e^{2} - 5 \, a^{2} c e^{3}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} A {\left | c \right |} - 7 \, {\left (2 \, \sqrt {a c} a c d^{2} e + a^{2} c d e^{2} - 3 \, \sqrt {a c} a^{2} e^{3}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d - \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, a^{3} c^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{2} \]________________________________________________________________________________________