\[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {\left (\sec ^{7}\left (d x +c \right )\right ) \left (b +a \sin \left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{7 d}+\frac {3 \left (\sec ^{5}\left (d x +c \right )\right ) \left (3 a b +\left (4 a^{2}-b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{70 d}-\frac {\left (\sec ^{3}\left (d x +c \right )\right ) \left (4 a b \left (a^{2}-b^{2}\right )-\left (32 a^{4}-39 a^{2} b^{2}+7 b^{4}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{140 \left (a^{2}-b^{2}\right ) d}-\frac {\sec \left (d x +c \right ) \left (a b \left (32 a^{4}-59 a^{2} b^{2}+27 b^{4}\right )-\left (128 a^{6}-272 a^{4} b^{2}+165 a^{2} b^{4}-21 b^{6}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{280 \left (a^{2}-b^{2}\right )^{2} d}+\frac {\left (128 a^{4}-144 a^{2} b^{2}+21 b^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{280 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \left (a^{2}-b^{2}\right ) d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}-\frac {2 a \left (8 a^{2}-3 b^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{35 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(sec(d*x+c)^8*(a+b*sin(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 (256 \, a^{5} - 384 \, a^{3} b^{2} + 123 \, a b^{4}\right )} \sqrt {i \, b} \cos \left (d x + c\right )^{7} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + \sqrt {2} {\left (256 \, a^{5} - 384 \, a^{3} b^{2} + 123 \, a b^{4}\right )} \sqrt {-i \, b} \cos \left (d x + c\right )^{7} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) + 3 \, \sqrt {2} {\left (128 i \, a^{4} b - 144 i \, a^{2} b^{3} + 21 i \, b^{5}\right )} \sqrt {i \, b} \cos \left (d x + c\right )^{7} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) + 3 \, \sqrt {2} {\left (-128 i \, a^{4} b + 144 i \, a^{2} b^{3} - 21 i \, b^{5}\right )} \sqrt {-i \, b} \cos \left (d x + c\right )^{7} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) - 6 \, {\left ({\left (32 \, a^{3} b^{2} - 27 \, a b^{4}\right )} \cos \left (d x + c\right )^{6} - 80 \, a^{3} b^{2} + 80 \, a b^{4} + 8 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (128 \, a^{4} b - 144 \, a^{2} b^{3} + 21 \, b^{5}\right )} \cos \left (d x + c\right )^{6} + 40 \, a^{4} b - 40 \, b^{5} + 2 \, {\left (32 \, a^{4} b - 39 \, a^{2} b^{3} + 7 \, b^{5}\right )} \cos \left (d x + c\right )^{4} + 12 \, {\left (4 \, a^{4} b - 5 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{1680 \, {\left (a^{2} b - b^{3}\right )} d \cos \left (d x + c\right )^{7}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (2 \, a b \sec \left (d x + c\right )^{8} \sin \left (d x + c\right ) - {\left (b^{2} \cos \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \sec \left (d x + c\right )^{8}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]