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