\[ \int \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (25 a^{4}-31 a^{2} b^{2}+6 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}}{105 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} d \sqrt {\cos \left (d x +c \right )}\, \sqrt {a +b \sec \left (d x +c \right )}}+\frac {16 b \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{35 d}+\frac {2 a \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{7 d}+\frac {2 \left (25 a^{2}+3 b^{2}\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{105 a d}+\frac {4 b \left (41 a^{2}-3 b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{105 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} d \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}} \]
command
integrate(cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {6 \, {\left (15 \, a^{4} \cos \left (d x + c\right )^{2} + 24 \, a^{3} b \cos \left (d x + c\right ) + 25 \, a^{4} + 3 \, a^{2} b^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-75 i \, a^{4} + 11 i \, a^{2} b^{2} - 12 i \, b^{4}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (75 i \, a^{4} - 11 i \, a^{2} b^{2} + 12 i \, b^{4}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 6 \, \sqrt {2} {\left (-41 i \, a^{3} b + 3 i \, a b^{3}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 6 \, \sqrt {2} {\left (41 i \, a^{3} b - 3 i \, a b^{3}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right )}{315 \, a^{3} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b \cos \left (d x + c\right )^{3} \sec \left (d x + c\right ) + a \cos \left (d x + c\right )^{3}\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}, x\right ) \]