\[ \int \frac {\sec ^2(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 b \sin \left (d x +c \right )}{7 d \left (b \cos \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {10 \sin \left (d x +c \right )}{21 b d \left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {10 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} d \sqrt {b \cos \left (d x +c \right )}} \]
command
integrate(sec(d*x+c)^2/(b*cos(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-5 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )} {\left (5 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right )}{21 \, b^{3} d \cos \left (d x + c\right )^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right )} \sec \left (d x + c\right )^{2}}{b^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]