\[ \int (b \cos (c+d x))^{5/2} \sec ^7(c+d x) \, dx \]
Optimal antiderivative \[ \frac {2 b^{6} \sin \left (d x +c \right )}{7 d \left (b \cos \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {10 b^{4} \sin \left (d x +c \right )}{21 d \left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {10 b^{3} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {b \cos \left (d x +c \right )}} \]
command
integrate((b*cos(d*x+c))^(5/2)*sec(d*x+c)^7,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-5 i \, \sqrt {2} b^{\frac {5}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} b^{\frac {5}{2}} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (5 \, b^{2} \cos \left (d x + c\right )^{2} + 3 \, b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{21 \, d \cos \left (d x + c\right )^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b \cos \left (d x + c\right )} b^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{7}, x\right ) \]