\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{a \left (a^{2}-b^{2}\right ) d \sqrt {a +b \sec \left (d x +c \right )}}-\frac {2 \left (48 A \,b^{3}-5 a^{3} B -40 B a \,b^{2}+6 a^{2} b \left (2 A +5 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} d \sqrt {\cos \left (d x +c \right )}\, \sqrt {a +b \sec \left (d x +c \right )}}-\frac {2 \left (6 A \,b^{2}-5 a b B -a^{2} \left (A -5 C \right )\right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{5 a^{2} \left (a^{2}-b^{2}\right ) d}+\frac {2 \left (24 A \,b^{3}+5 a^{3} B -20 B a \,b^{2}-a^{2} \left (9 A b -15 C b \right )\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{15 a^{3} \left (a^{2}-b^{2}\right ) d}-\frac {2 \left (48 A \,b^{4}+25 a^{3} b B -40 a \,b^{3} B -6 a^{2} b^{2} \left (4 A -5 C \right )-3 a^{4} \left (3 A +5 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} \left (a^{2}-b^{2}\right ) d \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}} \]
command
integrate(cos(d*x+c)^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {6 \, {\left (5 \, B a^{5} b - 3 \, {\left (3 \, A - 5 \, C\right )} a^{4} b^{2} - 20 \, B a^{3} b^{3} + 24 \, A a^{2} b^{4} + 3 \, {\left (A a^{6} - A a^{4} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (5 \, B a^{6} - 6 \, A a^{5} b - 5 \, B a^{4} b^{2} + 6 \, A a^{3} 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 )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - {\left (\sqrt {2} {\left (15 i \, B a^{6} - 3 i \, {\left (9 \, A + 25 \, C\right )} a^{5} b + 80 i \, B a^{4} b^{2} - 12 i \, {\left (7 \, A - 5 \, C\right )} a^{3} b^{3} - 80 i \, B a^{2} b^{4} + 96 i \, A a b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (15 i \, B a^{5} b - 3 i \, {\left (9 \, A + 25 \, C\right )} a^{4} b^{2} + 80 i \, B a^{3} b^{3} - 12 i \, {\left (7 \, A - 5 \, C\right )} a^{2} b^{4} - 80 i \, B a b^{5} + 96 i \, A b^{6}\right )}\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 ) - {\left (\sqrt {2} {\left (-15 i \, B a^{6} + 3 i \, {\left (9 \, A + 25 \, C\right )} a^{5} b - 80 i \, B a^{4} b^{2} + 12 i \, {\left (7 \, A - 5 \, C\right )} a^{3} b^{3} + 80 i \, B a^{2} b^{4} - 96 i \, A a b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-15 i \, B a^{5} b + 3 i \, {\left (9 \, A + 25 \, C\right )} a^{4} b^{2} - 80 i \, B a^{3} b^{3} + 12 i \, {\left (7 \, A - 5 \, C\right )} a^{2} b^{4} + 80 i \, B a b^{5} - 96 i \, A b^{6}\right )}\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 \, {\left (\sqrt {2} {\left (3 i \, {\left (3 \, A + 5 \, C\right )} a^{6} - 25 i \, B a^{5} b + 6 i \, {\left (4 \, A - 5 \, C\right )} a^{4} b^{2} + 40 i \, B a^{3} b^{3} - 48 i \, A a^{2} b^{4}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (3 i \, {\left (3 \, A + 5 \, C\right )} a^{5} b - 25 i \, B a^{4} b^{2} + 6 i \, {\left (4 \, A - 5 \, C\right )} a^{3} b^{3} + 40 i \, B a^{2} b^{4} - 48 i \, A a b^{5}\right )}\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 \, {\left (\sqrt {2} {\left (-3 i \, {\left (3 \, A + 5 \, C\right )} a^{6} + 25 i \, B a^{5} b - 6 i \, {\left (4 \, A - 5 \, C\right )} a^{4} b^{2} - 40 i \, B a^{3} b^{3} + 48 i \, A a^{2} b^{4}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-3 i \, {\left (3 \, A + 5 \, C\right )} a^{5} b + 25 i \, B a^{4} b^{2} - 6 i \, {\left (4 \, A - 5 \, C\right )} a^{3} b^{3} - 40 i \, B a^{2} b^{4} + 48 i \, A a b^{5}\right )}\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 )}{45 \, {\left ({\left (a^{8} - a^{6} b^{2}\right )} d \cos \left (d x + c\right ) + {\left (a^{7} b - a^{5} b^{3}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + B \cos \left (d x + c\right )^{2} \sec \left (d x + c\right ) + A \cos \left (d x + c\right )^{2}\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}, x\right ) \]