\[ \int \cos ^4(c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {8 \left (480 a^{4}-937 a^{2} b^{2}+231 b^{4}\right ) \cos \left (d x +c \right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{45045 b^{5} d}+\frac {8 a \left (40 a^{2}-81 b^{2}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3003 b^{4} d}-\frac {10 \left (16 a^{2}-33 b^{2}\right ) \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{1287 b^{3} d}+\frac {20 a \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{143 b^{2} d}-\frac {2 \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{13 b d}+\frac {16 a \left (160 a^{4}-279 a^{2} b^{2}+27 b^{4}\right ) \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}}{45045 b^{5} d}+\frac {8 \left (320 a^{6}-798 a^{4} b^{2}+435 a^{2} b^{4}-693 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{45045 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{6} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}-\frac {16 a \left (160 a^{6}-439 a^{4} b^{2}+306 a^{2} b^{4}-27 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{45045 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{6} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^4*sin(d*x+c)^2*(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, \sqrt {2} {\left (640 \, a^{7} - 1836 \, a^{5} b^{2} + 1401 \, a^{3} b^{4} + 531 \, 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 ) + 2 \, \sqrt {2} {\left (640 \, a^{7} - 1836 \, a^{5} b^{2} + 1401 \, a^{3} b^{4} + 531 \, 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 (-320 i \, a^{6} b + 798 i \, a^{4} b^{3} - 435 i \, a^{2} b^{5} + 693 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 (320 i \, a^{6} b - 798 i \, a^{4} b^{3} + 435 i \, a^{2} b^{5} - 693 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 (315 \, a b^{6} \cos \left (d x + c\right )^{5} - 5 \, {\left (80 \, a^{3} b^{4} - 57 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (160 \, a^{5} b^{2} - 279 \, a^{3} b^{4} + 27 \, a b^{6}\right )} \cos \left (d x + c\right ) + {\left (3465 \, b^{7} \cos \left (d x + c\right )^{5} + 35 \, {\left (10 \, a^{2} b^{5} - 33 \, b^{7}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (80 \, a^{4} b^{3} - 127 \, a^{2} b^{5} + 231 \, 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^{7} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (\cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{4}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]