\[ \int (a+b \cosh (x))^{3/2} (A+B \cosh (x)) \, dx \]
Optimal antiderivative \[ \frac {2 B \left (a +b \cosh \left (x \right )\right )^{\frac {3}{2}} \sinh \left (x \right )}{5}+\frac {2 \left (5 A b +3 B a \right ) \sinh \left (x \right ) \sqrt {a +b \cosh \left (x \right )}}{15}-\frac {2 i \left (20 A a b +3 B \,a^{2}+9 b^{2} B \right ) \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cosh \left (x \right )}}{15 \cosh \left (\frac {x}{2}\right ) b \sqrt {\frac {a +b \cosh \left (x \right )}{a +b}}}+\frac {2 i \left (a^{2}-b^{2}\right ) \left (5 A b +3 B a \right ) \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cosh \left (x \right )}{a +b}}}{15 \cosh \left (\frac {x}{2}\right ) b \sqrt {a +b \cosh \left (x \right )}} \]
command
integrate((a+b*cosh(x))^(3/2)*(A+B*cosh(x)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {4 \, {\left (\sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b - 18 \, B a b^{2} - 15 \, A b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b - 18 \, B a b^{2} - 15 \, A b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b - 18 \, B a b^{2} - 15 \, A b^{3}\right )} \sinh \left (x\right )^{2}\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 \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 12 \, {\left (\sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} + 9 \, B b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} + 9 \, B b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} + 9 \, B b^{3}\right )} \sinh \left (x\right )^{2}\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 \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (3 \, B b^{3} \cosh \left (x\right )^{4} + 3 \, B b^{3} \sinh \left (x\right )^{4} - 3 \, B b^{3} + 2 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (6 \, B b^{3} \cosh \left (x\right ) + 6 \, B a b^{2} + 5 \, A b^{3}\right )} \sinh \left (x\right )^{3} - 4 \, {\left (3 \, B a^{2} b + 20 \, A a b^{2} + 9 \, B b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (9 \, B b^{3} \cosh \left (x\right )^{2} - 6 \, B a^{2} b - 40 \, A a b^{2} - 18 \, B b^{3} + 3 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 2 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right ) + 2 \, {\left (6 \, B b^{3} \cosh \left (x\right )^{3} - 6 \, B a b^{2} - 5 \, A b^{3} + 3 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, B a^{2} b + 20 \, A a b^{2} + 9 \, B b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}}{90 \, {\left (b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B b \cosh \left (x\right )^{2} + A a + {\left (B a + A b\right )} \cosh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}, x\right ) \]