\[ \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ \frac {4 a^{2} \left (c -7 d \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{35 d f}-\frac {2 a^{2} \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{7 d f}+\frac {4 a^{2} \left (c^{2}-7 c d -10 d^{2}\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{35 d f}+\frac {4 a^{2} \left (c +3 d \right ) \left (c^{2}-10 c d -7 d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{35 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{2} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {4 a^{2} \left (c^{2}-7 c d -10 d^{2}\right ) \left (c^{2}-d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{35 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{2} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, \sqrt {2} {\left (a^{2} c^{4} - 7 \, a^{2} c^{3} d + 2 \, a^{2} c^{2} d^{2} + 21 \, a^{2} c d^{3} + 15 \, a^{2} d^{4}\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 ) + 2 \, \sqrt {2} {\left (a^{2} c^{4} - 7 \, a^{2} c^{3} d + 2 \, a^{2} c^{2} d^{2} + 21 \, a^{2} c d^{3} + 15 \, a^{2} d^{4}\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 (-i \, a^{2} c^{3} d + 7 i \, a^{2} c^{2} d^{2} + 37 i \, a^{2} c d^{3} + 21 i \, a^{2} d^{4}\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 (i \, a^{2} c^{3} d - 7 i \, a^{2} c^{2} d^{2} - 37 i \, a^{2} c d^{3} - 21 i \, a^{2} d^{4}\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 (5 \, a^{2} d^{4} \cos \left (f x + e\right )^{3} - 2 \, {\left (4 \, a^{2} c d^{3} + 7 \, a^{2} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a^{2} c^{2} d^{2} + 28 \, a^{2} c d^{3} + 25 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{105 \, d^{3} f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (2 \, a^{2} c + 2 \, a^{2} d - {\left (a^{2} c + 2 \, a^{2} d\right )} \cos \left (f x + e\right )^{2} - {\left (a^{2} d \cos \left (f x + e\right )^{2} - 2 \, a^{2} c - 2 \, a^{2} d\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}, x\right ) \]