\[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (a^{2}-b^{2}\right ) \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{a \,b^{2} d \sqrt {a +b \sin \left (d x +c \right )}}-\frac {8 \left (640 a^{4}-592 a^{2} b^{2}+15 b^{4}\right ) \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}}{1155 b^{6} d}+\frac {8 a \left (480 a^{2}-419 b^{2}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}}{1155 b^{5} d}-\frac {20 \left (32 a^{2}-27 b^{2}\right ) \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{231 b^{4} d}+\frac {2 \left (40 a^{2}-33 b^{2}\right ) \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{33 a \,b^{3} d}-\frac {2 \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{11 b^{2} d}+\frac {8 a \left (1280 a^{4}-1344 a^{2} b^{2}+123 b^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{1155 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{7} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}-\frac {8 \left (1280 a^{6}-1664 a^{4} b^{2}+369 a^{2} b^{4}+15 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{1155 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{7} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, {\left (\sqrt {2} {\left (2560 \, a^{6} b - 3648 \, a^{4} b^{3} + 984 \, a^{2} b^{5} + 45 \, b^{7}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (2560 \, a^{7} - 3648 \, a^{5} b^{2} + 984 \, a^{3} b^{4} + 45 \, a b^{6}\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 ) + 2 \, {\left (\sqrt {2} {\left (2560 \, a^{6} b - 3648 \, a^{4} b^{3} + 984 \, a^{2} b^{5} + 45 \, b^{7}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (2560 \, a^{7} - 3648 \, a^{5} b^{2} + 984 \, a^{3} b^{4} + 45 \, a b^{6}\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 (-1280 i \, a^{5} b^{2} + 1344 i \, a^{3} b^{4} - 123 i \, a b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-1280 i \, a^{6} b + 1344 i \, a^{4} b^{3} - 123 i \, a^{2} 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 (1280 i \, a^{5} b^{2} - 1344 i \, a^{3} b^{4} + 123 i \, a b^{6}\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (1280 i \, a^{6} b - 1344 i \, a^{4} b^{3} + 123 i \, a^{2} 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 (140 \, a b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (160 \, a^{3} b^{4} - 23 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (1280 \, a^{5} b^{2} - 1344 \, a^{3} b^{4} + 123 \, a b^{6}\right )} \cos \left (d x + c\right ) - {\left (105 \, b^{7} \cos \left (d x + c\right )^{5} - 5 \, {\left (40 \, a^{2} b^{5} + 3 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (320 \, a^{4} b^{3} - 246 \, a^{2} b^{5} - 15 \, b^{7}\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 )}}{3465 \, {\left (b^{9} d \sin \left (d x + c\right ) + a b^{8} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (\cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{4}\right )} \sqrt {b \sin \left (d x + c\right ) + a} \sin \left (d x + c\right )}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \]