\[ \int \cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {2 \left (a^{2}-b^{2}\right ) \left (16 A \,b^{3}-75 a^{3} B -24 B a \,b^{2}+6 a^{2} b \left (6 A +7 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d \sqrt {\cos \left (d x +c \right )}\, \sqrt {a +b \sec \left (d x +c \right )}}-\frac {2 \left (6 A \,b^{2}-9 a b B -7 a^{2} \left (7 A +9 C \right )\right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{315 a^{2} d}+\frac {2 \left (A b +9 B a \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{63 a d}+\frac {2 A \left (\cos ^{\frac {7}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{9 d}+\frac {2 \left (8 A \,b^{3}+75 a^{3} B -12 B a \,b^{2}+a^{2} b \left (13 A +21 C \right )\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{315 a^{3} d}-\frac {2 \left (16 A \,b^{4}-57 a^{3} b B -24 a \,b^{3} B +6 a^{2} b^{2} \left (4 A +7 C \right )-21 a^{4} \left (7 A +9 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{315 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}} \]
command
integrate(cos(d*x+c)^(9/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)*(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {6 \, {\left (35 \, A a^{5} \cos \left (d x + c\right )^{3} + 75 \, B a^{5} + {\left (13 \, A + 21 \, C\right )} a^{4} b - 12 \, B a^{3} b^{2} + 8 \, A a^{2} b^{3} + 5 \, {\left (9 \, B a^{5} + A a^{4} b\right )} \cos \left (d x + c\right )^{2} + {\left (7 \, {\left (7 \, A + 9 \, C\right )} a^{5} + 9 \, B a^{4} b - 6 \, A a^{3} b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-225 i \, B a^{5} - 3 i \, {\left (13 \, A + 21 \, C\right )} a^{4} b + 96 i \, B a^{3} b^{2} - 12 i \, {\left (3 \, A + 7 \, C\right )} a^{2} b^{3} + 48 i \, B a b^{4} - 32 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} + 3 i \, {\left (13 \, A + 21 \, C\right )} a^{4} b - 96 i \, B a^{3} b^{2} + 12 i \, {\left (3 \, A + 7 \, C\right )} a^{2} b^{3} - 48 i \, B a b^{4} + 32 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} - 57 i \, B a^{4} b + 6 i \, {\left (4 \, A + 7 \, C\right )} a^{3} b^{2} - 24 i \, B a^{2} b^{3} + 16 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} + 57 i \, B a^{4} b - 6 i \, {\left (4 \, A + 7 \, C\right )} a^{3} b^{2} + 24 i \, B a^{2} b^{3} - 16 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 )}{945 \, a^{5} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + B \cos \left (d x + c\right )^{4} \sec \left (d x + c\right ) + A \cos \left (d x + c\right )^{4}\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}, x\right ) \]