\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \sin \left (d x +c \right )}{3 a \left (a^{2}-b^{2}\right ) d \sec \left (d x +c \right )^{\frac {3}{2}} \left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2 \left (8 A \,b^{4}+9 a^{3} b B -5 a \,b^{3} B -2 a^{2} b^{2} \left (6 A -C \right )-6 a^{4} C \right ) \sin \left (d x +c \right )}{3 a^{2} \left (a^{2}-b^{2}\right )^{2} d \sec \left (d x +c \right )^{\frac {3}{2}} \sqrt {a +b \sec \left (d x +c \right )}}+\frac {2 \left (128 A \,b^{5}+5 B \,a^{5}+80 B \,a^{3} b^{2}-80 B a \,b^{4}-4 a^{2} b^{3} \left (29 A -10 C \right )-a^{4} b \left (17 A +45 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 )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{5} \left (a^{2}-b^{2}\right ) d \sqrt {a +b \sec \left (d x +c \right )}}+\frac {2 \left (48 A \,b^{4}+50 a^{3} b B -30 a \,b^{3} B +a^{4} \left (3 A -35 C \right )-a^{2} b^{2} \left (71 A -15 C \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{15 a^{3} \left (a^{2}-b^{2}\right )^{2} d \sec \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 \left (64 A \,b^{5}-5 B \,a^{5}+65 B \,a^{3} b^{2}-40 B a \,b^{4}+2 a^{4} b \left (7 A -20 C \right )-2 a^{2} b^{3} \left (49 A -10 C \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{15 a^{4} \left (a^{2}-b^{2}\right )^{2} d \sqrt {\sec \left (d x +c \right )}}+\frac {2 \left (128 A \,b^{6}-40 a^{5} b B +140 a^{3} b^{3} B -80 a \,b^{5} B +5 a^{4} b^{2} \left (11 A -15 C \right )-4 a^{2} b^{4} \left (53 A -10 C \right )+3 a^{6} \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 ) \sqrt {a +b \sec \left (d x +c \right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{5} \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)^(5/2)/(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
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\sec \left (d x + c\right )}}{b^{3} \sec \left (d x + c\right )^{6} + 3 \, a b^{2} \sec \left (d x + c\right )^{5} + 3 \, a^{2} b \sec \left (d x + c\right )^{4} + a^{3} \sec \left (d x + c\right )^{3}}, x\right ) \]