\[ \int \frac {(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {C \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{5 d \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {a^{\frac {3}{2}} \left (176 A +150 B +133 C \right ) \arcsinh \left (\frac {\sqrt {a}\, \tan \left (d x +c \right )}{\sqrt {a +a \sec \left (d x +c \right )}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{128 d}+\frac {a^{2} \left (80 A +90 B +67 C \right ) \sin \left (d x +c \right )}{240 d \cos \left (d x +c \right )^{\frac {7}{2}} \sqrt {a +a \sec \left (d x +c \right )}}+\frac {a^{2} \left (176 A +150 B +133 C \right ) \sin \left (d x +c \right )}{192 d \cos \left (d x +c \right )^{\frac {5}{2}} \sqrt {a +a \sec \left (d x +c \right )}}+\frac {a^{2} \left (176 A +150 B +133 C \right ) \sin \left (d x +c \right )}{128 d \cos \left (d x +c \right )^{\frac {3}{2}} \sqrt {a +a \sec \left (d x +c \right )}}+\frac {a \left (10 B +3 C \right ) \sin \left (d x +c \right ) \sqrt {a +a \sec \left (d x +c \right )}}{40 d \cos \left (d x +c \right )^{\frac {7}{2}}} \]
command
integrate((a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
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Maxima 5.44 via sagemath 9.3 output \[ \text {Timed out} \]_____________________________________________________