\[ \int \frac {\sqrt {\cos (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A -B \right ) \sin \left (d x +c \right )}{4 d \left (a +a \sec \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {\cos \left (d x +c \right )}}-\frac {\left (13 A -5 B \right ) \sin \left (d x +c \right )}{16 a d \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {\cos \left (d x +c \right )}}-\frac {\left (75 A -19 B \right ) \arctanh \left (\frac {\sin \left (d x +c \right ) \sqrt {a}\, \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {2}}{2 \sqrt {a +a \sec \left (d x +c \right )}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {2}}{32 a^{\frac {5}{2}} d}+\frac {\left (49 A -9 B \right ) \sin \left (d x +c \right )}{16 a^{2} d \sqrt {\cos \left (d x +c \right )}\, \sqrt {a +a \sec \left (d x +c \right )}} \]
command
integrate((A+B*sec(d*x+c))*cos(d*x+c)^(1/2)/(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 \[ \text {Timed out} \]_____________________________________________________