\[ \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {4 a^{3} \left (4 c^{3}-33 c^{2} d +182 c \,d^{2}+231 d^{3}\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{693 d^{2} f}-\frac {4 a^{3} \left (4 c^{2}-33 c d +189 d^{2}\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{693 d^{2} f}+\frac {8 a^{3} \left (c -6 d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{99 d^{2} f}-\frac {2 \cos \left (f x +e \right ) \left (a^{3}+a^{3} \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{11 d f}-\frac {4 a^{3} \left (4 c^{4}-33 c^{3} d +177 c^{2} d^{2}+561 c \,d^{3}+315 d^{4}\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{693 d^{2} f}-\frac {4 a^{3} \left (c +3 d \right ) \left (4 c^{4}-45 c^{3} d +309 c^{2} d^{2}+525 c \,d^{3}+231 d^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{693 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{3} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}+\frac {4 a^{3} \left (c^{2}-d^{2}\right ) \left (4 c^{4}-33 c^{3} d +177 c^{2} d^{2}+561 c \,d^{3}+315 d^{4}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{693 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{3} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^3*(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (\sqrt {2} {\left (8 \, a^{3} c^{6} - 66 \, a^{3} c^{5} d + 345 \, a^{3} c^{4} d^{2} + 330 \, a^{3} c^{3} d^{3} - 1392 \, a^{3} c^{2} d^{4} - 2376 \, a^{3} c d^{5} - 945 \, a^{3} d^{6}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + \sqrt {2} {\left (8 \, a^{3} c^{6} - 66 \, a^{3} c^{5} d + 345 \, a^{3} c^{4} d^{2} + 330 \, a^{3} c^{3} d^{3} - 1392 \, a^{3} c^{2} d^{4} - 2376 \, a^{3} c d^{5} - 945 \, a^{3} d^{6}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, \sqrt {2} {\left (4 i \, a^{3} c^{5} d - 33 i \, a^{3} c^{4} d^{2} + 174 i \, a^{3} c^{3} d^{3} + 1452 i \, a^{3} c^{2} d^{4} + 1806 i \, a^{3} c d^{5} + 693 i \, a^{3} d^{6}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, \sqrt {2} {\left (-4 i \, a^{3} c^{5} d + 33 i \, a^{3} c^{4} d^{2} - 174 i \, a^{3} c^{3} d^{3} - 1452 i \, a^{3} c^{2} d^{4} - 1806 i \, a^{3} c d^{5} - 693 i \, a^{3} d^{6}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (63 \, a^{3} d^{6} \cos \left (f x + e\right )^{5} - {\left (113 \, a^{3} c^{2} d^{4} + 627 \, a^{3} c d^{5} + 504 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )^{3} - {\left (4 \, a^{3} c^{4} d^{2} - 33 \, a^{3} c^{3} d^{3} - 1209 \, a^{3} c^{2} d^{4} - 2211 \, a^{3} c d^{5} - 1071 \, a^{3} d^{6}\right )} \cos \left (f x + e\right ) - {\left (7 \, {\left (23 \, a^{3} c d^{5} + 33 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{3} c^{3} d^{3} + 165 \, a^{3} c^{2} d^{4} + 427 \, a^{3} c d^{5} + 231 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{2079 \, d^{4} f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} + {\left (2 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (3 \, a^{3} c^{2} + 10 \, a^{3} c d + 7 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (a^{3} d^{2} \cos \left (f x + e\right )^{4} + 4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} - {\left (a^{3} c^{2} + 6 \, a^{3} c d + 5 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}, x\right ) \]