\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \sin \left (d x +c \right )}{a \left (a^{2}-b^{2}\right ) d \sqrt {\sec \left (d x +c \right )}\, \sqrt {a +b \sec \left (d x +c \right )}}+\frac {2 \left (8 A \,b^{2}-6 a b B +a^{2} \left (A +3 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}\, \left (\sqrt {\sec }\left (d x +c \right )\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} d \sqrt {a +b \sec \left (d x +c \right )}}-\frac {2 \left (4 A \,b^{2}-3 a b B -a^{2} \left (A -3 C \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{3 a^{2} \left (a^{2}-b^{2}\right ) d \sqrt {\sec \left (d x +c \right )}}+\frac {2 \left (8 A \,b^{3}+3 a^{3} B -6 B a \,b^{2}-a^{2} \left (5 A b -3 C b \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \sqrt {a +b \sec \left (d x +c \right )}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \left (a^{2}-b^{2}\right ) d \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}\, \sqrt {\sec \left (d x +c \right )}} \]
command
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2)/(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 {\sqrt {2} {\left (3 i \, {\left (A + 3 \, C\right )} a^{4} b - 15 i \, B a^{3} b^{2} + 2 i \, {\left (8 \, A - 3 \, C\right )} a^{2} b^{3} + 12 i \, B a b^{4} - 16 i \, A b^{5} + {\left (3 i \, {\left (A + 3 \, C\right )} a^{5} - 15 i \, B a^{4} b + 2 i \, {\left (8 \, A - 3 \, C\right )} a^{3} b^{2} + 12 i \, B a^{2} b^{3} - 16 i \, A a b^{4}\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 (A + 3 \, C\right )} a^{4} b + 15 i \, B a^{3} b^{2} - 2 i \, {\left (8 \, A - 3 \, C\right )} a^{2} b^{3} - 12 i \, B a b^{4} + 16 i \, A b^{5} + {\left (-3 i \, {\left (A + 3 \, C\right )} a^{5} + 15 i \, B a^{4} b - 2 i \, {\left (8 \, A - 3 \, C\right )} a^{3} b^{2} - 12 i \, B a^{2} b^{3} + 16 i \, A a b^{4}\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 - i \, {\left (5 \, A - 3 \, C\right )} a^{3} b^{2} - 6 i \, B a^{2} b^{3} + 8 i \, A a b^{4} + {\left (3 i \, B a^{5} - i \, {\left (5 \, A - 3 \, C\right )} a^{4} b - 6 i \, B a^{3} b^{2} + 8 i \, A a^{2} b^{3}\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 + i \, {\left (5 \, A - 3 \, C\right )} a^{3} b^{2} + 6 i \, B a^{2} b^{3} - 8 i \, A a b^{4} + {\left (-3 i \, B a^{5} + i \, {\left (5 \, A - 3 \, C\right )} a^{4} b + 6 i \, B a^{3} b^{2} - 8 i \, A a^{2} b^{3}\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 (A a^{5} - A a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (A - 3 \, C\right )} a^{4} b + 3 \, 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 )}}}{9 \, {\left ({\left (a^{7} - a^{5} b^{2}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b - a^{4} b^{3}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\sec \left (d x + c\right )}}{b^{2} \sec \left (d x + c\right )^{4} + 2 \, a b \sec \left (d x + c\right )^{3} + a^{2} \sec \left (d x + c\right )^{2}}, x\right ) \]