\[ \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 b^{2} \sin \left (d x +c \right )}{3 a \left (a^{2}-b^{2}\right ) d \left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {\sec \left (d x +c \right )}}+\frac {4 b^{2} \left (5 a^{2}-3 b^{2}\right ) \sin \left (d x +c \right )}{3 a^{2} \left (a^{2}-b^{2}\right )^{2} d \sqrt {\sec \left (d x +c \right )}\, \sqrt {a +b \sec \left (d x +c \right )}}+\frac {2 \left (a^{4}+16 a^{2} b^{2}-16 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}\, \left (\sqrt {\sec }\left (d x +c \right )\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} \left (a^{2}-b^{2}\right ) d \sqrt {a +b \sec \left (d x +c \right )}}+\frac {2 \left (a^{4}-13 a^{2} b^{2}+8 b^{4}\right ) \sin \left (d x +c \right ) \sqrt {a +b \sec \left (d x +c \right )}}{3 a^{3} \left (a^{2}-b^{2}\right )^{2} d \sqrt {\sec \left (d x +c \right )}}-\frac {8 b \left (2 a^{4}-7 a^{2} b^{2}+4 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {a}{a +b}}\right ) \sqrt {a +b \sec \left (d x +c \right )}}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} \left (a^{2}-b^{2}\right )^{2} d \sqrt {\frac {b +a \cos \left (d x +c \right )}{a +b}}\, \sqrt {\sec \left (d x +c \right )}} \]
command
integrate(1/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (-3 i \, a^{6} b^{2} - 37 i \, a^{4} b^{4} + 68 i \, a^{2} b^{6} - 32 i \, b^{8} + {\left (-3 i \, a^{8} - 37 i \, a^{6} b^{2} + 68 i \, a^{4} b^{4} - 32 i \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (3 i \, a^{7} b + 37 i \, a^{5} b^{3} - 68 i \, a^{3} b^{5} + 32 i \, a b^{7}\right )} \cos \left (d x + c\right )\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 (3 i \, a^{6} b^{2} + 37 i \, a^{4} b^{4} - 68 i \, a^{2} b^{6} + 32 i \, b^{8} + {\left (3 i \, a^{8} + 37 i \, a^{6} b^{2} - 68 i \, a^{4} b^{4} + 32 i \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (-3 i \, a^{7} b - 37 i \, a^{5} b^{3} + 68 i \, a^{3} b^{5} - 32 i \, a b^{7}\right )} \cos \left (d x + c\right )\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 ) - 12 \, \sqrt {2} {\left (2 i \, a^{5} b^{3} - 7 i \, a^{3} b^{5} + 4 i \, a b^{7} + {\left (2 i \, a^{7} b - 7 i \, a^{5} b^{3} + 4 i \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 i \, a^{6} b^{2} - 7 i \, a^{4} b^{4} + 4 i \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\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 ) - 12 \, \sqrt {2} {\left (-2 i \, a^{5} b^{3} + 7 i \, a^{3} b^{5} - 4 i \, a b^{7} + {\left (-2 i \, a^{7} b + 7 i \, a^{5} b^{3} - 4 i \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (-2 i \, a^{6} b^{2} + 7 i \, a^{4} b^{4} - 4 i \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\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 ) + \frac {6 \, {\left ({\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (a^{7} b - 8 \, a^{5} b^{3} + 5 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{6} b^{2} - 13 \, a^{4} b^{4} + 8 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{9 \, {\left ({\left (a^{11} - 2 \, a^{9} b^{2} + a^{7} b^{4}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b - 2 \, a^{8} b^{3} + a^{6} b^{5}\right )} d \cos \left (d x + c\right ) + {\left (a^{9} b^{2} - 2 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\sec \left (d x + c\right )}}{b^{3} \sec \left (d x + c\right )^{5} + 3 \, a b^{2} \sec \left (d x + c\right )^{4} + 3 \, a^{2} b \sec \left (d x + c\right )^{3} + a^{3} \sec \left (d x + c\right )^{2}}, x\right ) \]