\[ \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A -B \right ) \sin \left (d x +c \right )}{4 d \sec \left (d x +c \right )^{\frac {3}{2}} \left (a +a \sec \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {\left (21 A -13 B \right ) \sin \left (d x +c \right )}{16 a d \sec \left (d x +c \right )^{\frac {3}{2}} \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\left (283 A -163 B \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {2}}{2 \sqrt {a +a \sec \left (d x +c \right )}}\right ) \sqrt {2}}{32 a^{\frac {5}{2}} d}+\frac {\left (157 A -85 B \right ) \sin \left (d x +c \right )}{80 a^{2} d \sec \left (d x +c \right )^{\frac {3}{2}} \sqrt {a +a \sec \left (d x +c \right )}}-\frac {\left (787 A -475 B \right ) \sin \left (d x +c \right )}{240 a^{2} d \sqrt {\sec \left (d x +c \right )}\, \sqrt {a +a \sec \left (d x +c \right )}}+\frac {\left (2671 A -1495 B \right ) \sin \left (d x +c \right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{240 a^{2} d \sqrt {a +a \sec \left (d x +c \right )}} \]
command
integrate((A+B*sec(d*x+c))/sec(d*x+c)^(5/2)/(a+a*sec(d*x+c))^(5/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \text {output too large to display} \]
Maxima 5.44 via sagemath 9.3 output \[ \text {Timed out} \]_____________________________________________________