\[ \int (a+b \cos (c+d x))^{3/2} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {4 a b \left (a +b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{35 d}-\frac {2 b \left (a +b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{7 d}+\frac {2 b \left (41 a^{2}-25 b^{2}\right ) \sin \left (d x +c \right ) \sqrt {a +b \cos \left (d x +c \right )}}{105 d}+\frac {4 a \left (73 a^{2}-41 b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cos \left (d x +c \right )}}{105 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}-\frac {2 \left (41 a^{4}-66 a^{2} b^{2}+25 b^{4}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cos \left (d x +c \right )}{a +b}}}{105 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {a +b \cos \left (d x +c \right )}} \]
command
integrate((a+b*cos(d*x+c))^(3/2)*(a^2-b^2*cos(d*x+c)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (-23 i \, a^{4} - 116 i \, a^{2} b^{2} + 75 i \, b^{4}\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 (23 i \, a^{4} + 116 i \, a^{2} b^{2} - 75 i \, b^{4}\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 ) - 6 \, \sqrt {2} {\left (-73 i \, a^{3} b + 41 i \, a b^{3}\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 \, \sqrt {2} {\left (73 i \, a^{3} b - 41 i \, a b^{3}\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 (15 \, b^{4} \cos \left (d x + c\right )^{2} + 24 \, a b^{3} \cos \left (d x + c\right ) - 32 \, a^{2} b^{2} + 25 \, b^{4}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, b d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (b^{3} \cos \left (d x + c\right )^{3} + a b^{2} \cos \left (d x + c\right )^{2} - a^{2} b \cos \left (d x + c\right ) - a^{3}\right )} \sqrt {b \cos \left (d x + c\right ) + a}, x\right ) \]