\[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 a \left (\cos ^{5}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{39 d}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{15 d}-\frac {2 \left (3 a^{2}+13 b^{2}\right ) \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{429 d}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (8 a^{4}-33 a^{2} b^{2}-39 b^{4}-7 a b \left (a^{2}+63 b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{9009 b^{2} d}+\frac {4 \cos \left (d x +c \right ) \left (32 a^{6}-165 a^{4} b^{2}+450 a^{2} b^{4}+195 b^{6}-24 a b \left (a^{4}-5 a^{2} b^{2}-60 b^{4}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{45045 b^{4} d}-\frac {8 a \left (32 a^{6}-189 a^{4} b^{2}+570 a^{2} b^{4}+1635 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{45045 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{5} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}+\frac {8 \left (32 a^{8}-197 a^{6} b^{2}+615 a^{4} b^{4}-255 a^{2} b^{6}-195 b^{8}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{45045 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{5} d \sqrt {a +b \sin \left (d x +c \right )}} \]
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 (2 \, \sqrt {2} {\left (64 \, a^{8} - 402 \, a^{6} b^{2} + 1275 \, a^{4} b^{4} - 2400 \, a^{2} b^{6} - 585 \, b^{8}\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 \, \sqrt {2} {\left (64 \, a^{8} - 402 \, a^{6} b^{2} + 1275 \, a^{4} b^{4} - 2400 \, a^{2} b^{6} - 585 \, b^{8}\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^{7} b - 189 i \, a^{5} b^{3} + 570 i \, a^{3} b^{5} + 1635 i \, a 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^{7} b + 189 i \, a^{5} b^{3} - 570 i \, a^{3} b^{5} - 1635 i \, a 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 (3003 \, b^{8} \cos \left (d x + c\right )^{7} - 21 \, {\left (213 \, a^{2} b^{6} + 208 \, b^{8}\right )} \cos \left (d x + c\right )^{5} - 5 \, {\left (8 \, a^{4} b^{4} - 33 \, a^{2} b^{6} - 39 \, b^{8}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (32 \, a^{6} b^{2} - 165 \, a^{4} b^{4} + 450 \, a^{2} b^{6} + 195 \, b^{8}\right )} \cos \left (d x + c\right ) - {\left (7161 \, a b^{7} \cos \left (d x + c\right )^{5} - 35 \, {\left (a^{3} b^{5} + 63 \, a b^{7}\right )} \cos \left (d x + c\right )^{3} + 48 \, {\left (a^{5} b^{3} - 5 \, a^{3} b^{5} - 60 \, a 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 )}}{135135 \, b^{6} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (2 \, a b \cos \left (d x + c\right )^{6} - 2 \, a b \cos \left (d x + c\right )^{4} + {\left (b^{2} \cos \left (d x + c\right )^{6} - {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]