\[ \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ \frac {4 b \left (-7 a d +b c \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{35 d f}-\frac {2 b^{2} \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{7 d f}-\frac {2 \left (5 \left (7 a^{2}+5 b^{2}\right ) d^{2}-6 b c \left (-7 a d +b c \right )\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{105 d f}-\frac {4 \left (70 a^{2} c \,d^{2}+21 a b d \left (c^{2}+3 d^{2}\right )-b^{2} \left (3 c^{3}-41 c \,d^{2}\right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{105 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{2} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}+\frac {2 \left (c^{2}-d^{2}\right ) \left (42 a b c d +35 a^{2} d^{2}-b^{2} \left (6 c^{2}-25 d^{2}\right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{105 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{2} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+b*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 {\sqrt {2} {\left (12 \, b^{2} c^{4} - 84 \, a b c^{3} d + 252 \, a b c d^{3} + {\left (35 \, a^{2} - 11 \, b^{2}\right )} c^{2} d^{2} + 15 \, {\left (7 \, a^{2} + 5 \, b^{2}\right )} 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 ) + \sqrt {2} {\left (12 \, b^{2} c^{4} - 84 \, a b c^{3} d + 252 \, a b c d^{3} + {\left (35 \, a^{2} - 11 \, b^{2}\right )} c^{2} d^{2} + 15 \, {\left (7 \, a^{2} + 5 \, b^{2}\right )} 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 ) - 6 \, \sqrt {2} {\left (-3 i \, b^{2} c^{3} d + 21 i \, a b c^{2} d^{2} + 63 i \, a b d^{4} + i \, {\left (70 \, a^{2} + 41 \, b^{2}\right )} c d^{3}\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 ) - 6 \, \sqrt {2} {\left (3 i \, b^{2} c^{3} d - 21 i \, a b c^{2} d^{2} - 63 i \, a b d^{4} - i \, {\left (70 \, a^{2} + 41 \, b^{2}\right )} c d^{3}\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 ) + 6 \, {\left (15 \, b^{2} d^{4} \cos \left (f x + e\right )^{3} - 6 \, {\left (4 \, b^{2} c d^{3} + 7 \, a b d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (3 \, b^{2} c^{2} d^{2} + 84 \, a b c d^{3} + 5 \, {\left (7 \, a^{2} + 8 \, b^{2}\right )} d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{315 \, d^{3} f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (2 \, a b d - {\left (b^{2} c + 2 \, a b d\right )} \cos \left (f x + e\right )^{2} + {\left (a^{2} + b^{2}\right )} c - {\left (b^{2} d \cos \left (f x + e\right )^{2} - 2 \, a b c - {\left (a^{2} + b^{2}\right )} d\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}, x\right ) \]