\[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 a \left (8 a^{2}+67 b^{2}\right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{693 b^{2} d}+\frac {2 \left (8 a^{2}+81 b^{2}\right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{693 b^{2} d}-\frac {8 a \left (a +b \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{99 b^{2} d}+\frac {2 \cos \left (d x +c \right ) \left (a +b \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{11 b d}+\frac {2 \left (8 a^{4}+57 a^{2} b^{2}+135 b^{4}\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{693 b^{2} d}+\frac {2 a \left (8 a^{4}+51 a^{2} b^{2}+741 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{693 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}-\frac {2 \left (8 a^{6}+49 a^{4} b^{2}+78 a^{2} b^{4}-135 b^{6}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{693 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3} d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (16 i \, a^{6} + 96 i \, a^{4} b^{2} - 507 i \, a^{2} b^{4} - 405 i \, b^{6}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right ) + \sqrt {2} {\left (-16 i \, a^{6} - 96 i \, a^{4} b^{2} + 507 i \, a^{2} b^{4} + 405 i \, b^{6}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-8 i \, a^{5} b - 51 i \, a^{3} b^{3} - 741 i \, a b^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (8 i \, a^{5} b + 51 i \, a^{3} b^{3} + 741 i \, a b^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, 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 \, a^{3} - 9 \, 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 \, a}{3 \, b}\right )\right ) + 6 \, {\left (63 \, b^{6} \cos \left (d x + c\right )^{4} + 161 \, a b^{5} \cos \left (d x + c\right )^{3} - 4 \, a^{4} b^{2} + 205 \, a^{2} b^{4} + 135 \, b^{6} + {\left (113 \, a^{2} b^{4} + 81 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, a^{3} b^{3} + 229 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{2079 \, b^{4} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{2} \cos \left (d x + c\right )^{5} + 2 \, a b \cos \left (d x + c\right )^{4} + a^{2} \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]