\[ \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (54 a b c d -189 a^{2} d^{2}-b^{2} \left (8 c^{2}+49 d^{2}\right )\right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{315 d^{2} f}+\frac {8 b^{2} \left (-5 a d +b c \right ) \cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{63 d^{2} f}-\frac {2 b^{2} \cos \left (f x +e \right ) \left (a +b \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{9 d f}-\frac {2 \left (189 a^{2} b c \,d^{2}+105 a^{3} d^{3}-9 a \,b^{2} d \left (6 c^{2}-25 d^{2}\right )+b^{3} \left (8 c^{3}+39 c \,d^{2}\right )\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{315 d^{2} f}-\frac {2 \left (420 a^{3} c \,d^{3}+189 a^{2} b \,d^{2} \left (c^{2}+3 d^{2}\right )-a \,b^{2} \left (54 c^{3} d -738 c \,d^{3}\right )+b^{3} \left (8 c^{4}+33 c^{2} d^{2}+147 d^{4}\right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{315 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{3} f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}+\frac {2 \left (c^{2}-d^{2}\right ) \left (189 a^{2} b c \,d^{2}+105 a^{3} d^{3}-9 a \,b^{2} d \left (6 c^{2}-25 d^{2}\right )+b^{3} \left (8 c^{3}+39 c \,d^{2}\right )\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{315 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) d^{3} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((a+b*sin(f*x+e))^3*(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 (16 \, b^{3} c^{5} - 108 \, a b^{2} c^{4} d + 6 \, {\left (63 \, a^{2} b + 10 \, b^{3}\right )} c^{3} d^{2} - 3 \, {\left (35 \, a^{3} - 33 \, a b^{2}\right )} c^{2} d^{3} - 6 \, {\left (189 \, a^{2} b + 44 \, b^{3}\right )} c d^{4} - 45 \, {\left (7 \, a^{3} + 15 \, a b^{2}\right )} d^{5}\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 (16 \, b^{3} c^{5} - 108 \, a b^{2} c^{4} d + 6 \, {\left (63 \, a^{2} b + 10 \, b^{3}\right )} c^{3} d^{2} - 3 \, {\left (35 \, a^{3} - 33 \, a b^{2}\right )} c^{2} d^{3} - 6 \, {\left (189 \, a^{2} b + 44 \, b^{3}\right )} c d^{4} - 45 \, {\left (7 \, a^{3} + 15 \, a b^{2}\right )} d^{5}\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 (8 i \, b^{3} c^{4} d - 54 i \, a b^{2} c^{3} d^{2} + 3 i \, {\left (63 \, a^{2} b + 11 \, b^{3}\right )} c^{2} d^{3} + 6 i \, {\left (70 \, a^{3} + 123 \, a b^{2}\right )} c d^{4} + 21 i \, {\left (27 \, a^{2} b + 7 \, b^{3}\right )} d^{5}\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 (-8 i \, b^{3} c^{4} d + 54 i \, a b^{2} c^{3} d^{2} - 3 i \, {\left (63 \, a^{2} b + 11 \, b^{3}\right )} c^{2} d^{3} - 6 i \, {\left (70 \, a^{3} + 123 \, a b^{2}\right )} c d^{4} - 21 i \, {\left (27 \, a^{2} b + 7 \, b^{3}\right )} d^{5}\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 (5 \, {\left (10 \, b^{3} c d^{4} + 27 \, a b^{2} d^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (4 \, b^{3} c^{3} d^{2} - 27 \, a b^{2} c^{2} d^{3} - 6 \, {\left (63 \, a^{2} b + 23 \, b^{3}\right )} c d^{4} - 15 \, {\left (7 \, a^{3} + 24 \, a b^{2}\right )} d^{5}\right )} \cos \left (f x + e\right ) + {\left (35 \, b^{3} d^{5} \cos \left (f x + e\right )^{3} - 3 \, {\left (b^{3} c^{2} d^{3} + 72 \, a b^{2} c d^{4} + 7 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{945 \, d^{4} f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{3} d \cos \left (f x + e\right )^{4} - {\left (3 \, a b^{2} c + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} d\right )} \cos \left (f x + e\right )^{2} + {\left (a^{3} + 3 \, a b^{2}\right )} c + {\left (3 \, a^{2} b + b^{3}\right )} d - {\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} \cos \left (f x + e\right )^{2} - {\left (3 \, a^{2} b + b^{3}\right )} c - {\left (a^{3} + 3 \, a b^{2}\right )} d\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}, x\right ) \]