\[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 b \left (\cos ^{3}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{9 d}-\frac {8 a b \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{21 d}+\frac {2 \cos \left (d x +c \right ) \left (a \left (5 a^{2}+27 b^{2}\right )+3 b \left (25 a^{2}+7 b^{2}\right ) \sin \left (d x +c \right )\right ) \sqrt {a +b \sin \left (d x +c \right )}}{315 b d}-\frac {4 \left (5 a^{4}+102 a^{2} b^{2}+21 b^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \sin \left (d x +c \right )}}{315 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{2} d \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}+\frac {4 a \left (5 a^{4}+22 a^{2} b^{2}-27 b^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \sin \left (d x +c \right )}{a +b}}}{315 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) b^{2} d \sqrt {a +b \sin \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^2*(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 (5 \, a^{5} - 18 \, a^{3} b^{2} - 51 \, a b^{4}\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 (5 \, a^{5} - 18 \, a^{3} b^{2} - 51 \, a b^{4}\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 ) + 3 \, \sqrt {2} {\left (5 i \, a^{4} b + 102 i \, a^{2} b^{3} + 21 i \, b^{5}\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 \, \sqrt {2} {\left (-5 i \, a^{4} b - 102 i \, a^{2} b^{3} - 21 i \, b^{5}\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 (95 \, a b^{4} \cos \left (d x + c\right )^{3} - {\left (5 \, a^{3} b^{2} + 27 \, a b^{4}\right )} \cos \left (d x + c\right ) + {\left (35 \, b^{5} \cos \left (d x + c\right )^{3} - 3 \, {\left (25 \, a^{2} b^{3} + 7 \, b^{5}\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 )}}{945 \, b^{3} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (b^{2} \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]