\[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 b \left (\cos ^{5}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{13 d}-\frac {32 a b \left (\cos ^{5}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{143 d}+\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (a \left (5 a^{2}+59 b^{2}\right )+7 b \left (53 a^{2}+11 b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{3003 b d}-\frac {4 \cos \left (d x +c \right ) \left (4 a \left (5 a^{4}-40 a^{2} b^{2}-93 b^{4}\right )-3 b \left (5 a^{4}+430 a^{2} b^{2}+77 b^{4}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{15015 b^{3} d}+\frac {8 \left (20 a^{6}-175 a^{4} b^{2}-1662 a^{2} b^{4}-231 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{15015 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{4} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}-\frac {32 a \left (5 a^{6}-45 a^{4} b^{2}-53 a^{2} b^{4}+93 b^{6}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{15015 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{4} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^4*(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 (40 \, a^{7} - 365 \, a^{5} b^{2} + 1026 \, a^{3} b^{4} + 1347 \, 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 (40 \, a^{7} - 365 \, a^{5} b^{2} + 1026 \, a^{3} b^{4} + 1347 \, 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 (-20 i \, a^{6} b + 175 i \, a^{4} b^{3} + 1662 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 (20 i \, a^{6} b - 175 i \, a^{4} b^{3} - 1662 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 (2835 \, a b^{6} \cos \left (d x + c\right )^{5} - 5 \, {\left (5 \, a^{3} b^{4} + 59 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, a^{5} b^{2} - 40 \, a^{3} b^{4} - 93 \, a b^{6}\right )} \cos \left (d x + c\right ) + {\left (1155 \, b^{7} \cos \left (d x + c\right )^{5} - 35 \, {\left (53 \, a^{2} b^{5} + 11 \, b^{7}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (5 \, a^{4} b^{3} + 430 \, 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^{5} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (b^{2} \cos \left (d x + c\right )^{6} - 2 \, a b \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{4}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]