\[ \int \frac {(a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 A \left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{9 d \sec \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 \left (a^{2}-b^{2}\right ) \left (8 A \,b^{3}+75 a^{3} B -18 B a \,b^{2}+a^{2} \left (39 A b +63 C b \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}\, \left (\sqrt {\sec }\left (d x +c \right )\right )}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d \sqrt {a +b \sec \left (d x +c \right )}}+\frac {2 \left (A b +3 B a \right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{21 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (3 A \,b^{2}+72 a b B +7 a^{2} \left (7 A +9 C \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{315 a d \sec \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (4 A \,b^{3}-75 a^{3} B -9 B a \,b^{2}-2 a^{2} b \left (44 A +63 C \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{315 a^{2} d \sqrt {\sec \left (d x +c \right )}}+\frac {2 \left (8 A \,b^{4}+246 a^{3} b B -18 a \,b^{3} B +21 a^{4} \left (7 A +9 C \right )+3 a^{2} b^{2} \left (11 A +21 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \sqrt {a +b \sec \left (d x +c \right )}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}\, \sqrt {\sec \left (d x +c \right )}} \]
command
integrate((a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (-225 i \, B a^{5} - 6 i \, {\left (44 \, A + 63 \, C\right )} a^{4} b + 33 i \, B a^{3} b^{2} + 6 i \, {\left (10 \, A + 21 \, C\right )} a^{2} b^{3} - 36 i \, B a b^{4} + 16 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (225 i \, B a^{5} + 6 i \, {\left (44 \, A + 63 \, C\right )} a^{4} b - 33 i \, B a^{3} b^{2} - 6 i \, {\left (10 \, A + 21 \, C\right )} a^{2} b^{3} + 36 i \, B a b^{4} - 16 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-21 i \, {\left (7 \, A + 9 \, C\right )} a^{5} - 246 i \, B a^{4} b - 3 i \, {\left (11 \, A + 21 \, C\right )} a^{3} b^{2} + 18 i \, B a^{2} b^{3} - 8 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (21 i \, {\left (7 \, A + 9 \, C\right )} a^{5} + 246 i \, B a^{4} b + 3 i \, {\left (11 \, A + 21 \, C\right )} a^{3} b^{2} - 18 i \, B a^{2} b^{3} + 8 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) + \frac {6 \, {\left (35 \, A a^{5} \cos \left (d x + c\right )^{4} + 5 \, {\left (9 \, B a^{5} + 10 \, A a^{4} b\right )} \cos \left (d x + c\right )^{3} + {\left (7 \, {\left (7 \, A + 9 \, C\right )} a^{5} + 72 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (75 \, B a^{5} + 2 \, {\left (44 \, A + 63 \, C\right )} a^{4} b + 9 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{945 \, a^{4} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C b \sec \left (d x + c\right )^{3} + {\left (C a + B b\right )} \sec \left (d x + c\right )^{2} + A a + {\left (B a + A b\right )} \sec \left (d x + c\right )\right )} \sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]