\[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (8 a +3 b \sin \left (d x +c \right )\right )}{15 b^{2} d \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {4 \cos \left (d x +c \right ) \left (32 a^{2}-9 b^{2}+8 a b \sin \left (d x +c \right )\right )}{15 b^{4} d \sqrt {a +b \sin \left (d x +c \right )}}-\frac {8 \left (32 a^{2}-9 b^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{15 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{5} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}+\frac {8 a \left (32 a^{2}-17 b^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{15 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{5} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^4*sin(d*x+c)/(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 {2 \, {\left (4 \, {\left (\sqrt {2} {\left (32 \, a^{3} b^{2} - 21 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (32 \, a^{4} b - 21 \, a^{2} b^{3}\right )} \sin \left (d x + c\right ) - \sqrt {2} {\left (32 \, a^{5} + 11 \, a^{3} b^{2} - 21 \, a b^{4}\right )}\right )} \sqrt {i \, b} {\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 ) + 4 \, {\left (\sqrt {2} {\left (32 \, a^{3} b^{2} - 21 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} {\left (32 \, a^{4} b - 21 \, a^{2} b^{3}\right )} \sin \left (d x + c\right ) - \sqrt {2} {\left (32 \, a^{5} + 11 \, a^{3} b^{2} - 21 \, a b^{4}\right )}\right )} \sqrt {-i \, b} {\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 ) - 6 \, {\left (\sqrt {2} {\left (-32 i \, a^{2} b^{3} + 9 i \, b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (32 i \, a^{3} b^{2} - 9 i \, a b^{4}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (32 i \, a^{4} b + 23 i \, a^{2} b^{3} - 9 i \, b^{5}\right )}\right )} \sqrt {i \, b} {\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 (\sqrt {2} {\left (32 i \, a^{2} b^{3} - 9 i \, b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-32 i \, a^{3} b^{2} + 9 i \, a b^{4}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-32 i \, a^{4} b - 23 i \, a^{2} b^{3} + 9 i \, b^{5}\right )}\right )} \sqrt {-i \, b} {\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 \, {\left (8 \, a b^{4} \cos \left (d x + c\right )^{3} - 2 \, {\left (32 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right ) - {\left (3 \, b^{5} \cos \left (d x + c\right )^{3} + 2 \, {\left (40 \, a^{2} b^{3} - 9 \, b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}\right )}}{45 \, {\left (b^{8} d \cos \left (d x + c\right )^{2} - 2 \, a b^{7} d \sin \left (d x + c\right ) - {\left (a^{2} b^{6} + b^{8}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )}{3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]