\[ \int \frac {\sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \sin \left (d x +c \right )}{5 d \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {6 \sin \left (d x +c \right )}{5 b^{2} d \sqrt {b \cos \left (d x +c \right )}}-\frac {6 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {b \cos \left (d x +c \right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate(sec(d*x+c)/(b*cos(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-3 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )} {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right )}{5 \, b^{3} d \cos \left (d x + c\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right )} \sec \left (d x + c\right )}{b^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]