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