\[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (\cos ^{5}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{13 d}-\frac {6 a \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{143 d}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (4 a \left (2 a^{2}-5 b^{2}\right )-7 b \left (a^{2}+11 b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{3003 b^{2} d}+\frac {4 \cos \left (d x +c \right ) \left (a \left (32 a^{4}-113 a^{2} b^{2}+177 b^{4}\right )-3 b \left (8 a^{4}-27 a^{2} b^{2}-77 b^{4}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{15015 b^{4} d}-\frac {8 \left (32 a^{6}-137 a^{4} b^{2}+258 a^{2} b^{4}+231 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{15015 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{5} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}+\frac {8 a \left (32 a^{6}-145 a^{4} b^{2}+290 a^{2} b^{4}-177 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{15015 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{5} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (4 \, \sqrt {2} {\left (32 \, a^{7} - 149 \, a^{5} b^{2} + 306 \, a^{3} b^{4} - 381 \, a b^{6}\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 \, \sqrt {2} {\left (32 \, a^{7} - 149 \, a^{5} b^{2} + 306 \, a^{3} b^{4} - 381 \, a b^{6}\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 \, \sqrt {2} {\left (32 i \, a^{6} b - 137 i \, a^{4} b^{3} + 258 i \, a^{2} b^{5} + 231 i \, b^{7}\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 \, \sqrt {2} {\left (-32 i \, a^{6} b + 137 i \, a^{4} b^{3} - 258 i \, a^{2} b^{5} - 231 i \, b^{7}\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 (1470 \, a b^{6} \cos \left (d x + c\right )^{5} + 20 \, {\left (2 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (32 \, a^{5} b^{2} - 113 \, a^{3} b^{4} + 177 \, a b^{6}\right )} \cos \left (d x + c\right ) + {\left (1155 \, b^{7} \cos \left (d x + c\right )^{5} - 35 \, {\left (a^{2} b^{5} + 11 \, b^{7}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (8 \, a^{4} b^{3} - 27 \, a^{2} b^{5} - 77 \, 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 )}}{45045 \, b^{6} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (b \cos \left (d x + c\right )^{6} - a \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - b \cos \left (d x + c\right )^{4}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]