\[ \int \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (a^{2}-b^{2}\right ) \left (25 a^{2} A +8 A \,b^{2}-14 a b B +35 a^{2} C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}}{105 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d \sqrt {\cos \left (d x +c \right )}\, \sqrt {a +b \sec \left (d x +c \right )}}+\frac {2 \left (A b +7 B a \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{35 a d}+\frac {2 A \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{7 d}-\frac {2 \left (4 A \,b^{2}-7 a b B -5 a^{2} \left (5 A +7 C \right )\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{105 a^{2} d}+\frac {2 \left (8 A \,b^{3}+63 a^{3} B -14 B a \,b^{2}+a^{2} b \left (19 A +35 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{105 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}} \]
command
integrate(cos(d*x+c)^(7/2)*(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 (15 \, A a^{4} \cos \left (d x + c\right )^{2} + 5 \, {\left (5 \, A + 7 \, C\right )} a^{4} + 7 \, B a^{3} b - 4 \, A a^{2} b^{2} + 3 \, {\left (7 \, B a^{4} + A a^{3} b\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 (-15 i \, {\left (5 \, A + 7 \, C\right )} a^{4} - 21 i \, B a^{3} b + 2 i \, {\left (16 \, A + 35 \, C\right )} a^{2} b^{2} - 28 i \, B a b^{3} + 16 i \, A b^{4}\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 (15 i \, {\left (5 \, A + 7 \, C\right )} a^{4} + 21 i \, B a^{3} b - 2 i \, {\left (16 \, A + 35 \, C\right )} a^{2} b^{2} + 28 i \, B a b^{3} - 16 i \, A b^{4}\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 (-63 i \, B a^{4} - i \, {\left (19 \, A + 35 \, C\right )} a^{3} b + 14 i \, B a^{2} b^{2} - 8 i \, A a b^{3}\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 (63 i \, B a^{4} + i \, {\left (19 \, A + 35 \, C\right )} a^{3} b - 14 i \, B a^{2} b^{2} + 8 i \, A a b^{3}\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 )}{315 \, a^{4} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{3} \sec \left (d x + c\right )^{2} + B \cos \left (d x + c\right )^{3} \sec \left (d x + c\right ) + A \cos \left (d x + c\right )^{3}\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}, x\right ) \]