\[ \int (a+b \sinh (x))^{5/2} (A+B \sinh (x)) \, dx \]
Optimal antiderivative \[ \frac {2 \left (7 A b +5 B a \right ) \cosh \left (x \right ) \left (a +b \sinh \left (x \right )\right )^{\frac {3}{2}}}{35}+\frac {2 B \cosh \left (x \right ) \left (a +b \sinh \left (x \right )\right )^{\frac {5}{2}}}{7}+\frac {2 \left (56 A a b +15 B \,a^{2}-25 b^{2} B \right ) \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}{105}+\frac {2 i \left (161 A \,a^{2} b -63 A \,b^{3}+15 a^{3} B -145 B a \,b^{2}\right ) \sqrt {\frac {1}{2}+\frac {i \sinh \left (x \right )}{2}}\, \EllipticE \left (\cos \left (\frac {\pi }{4}+\frac {i x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{i a +b}}\right ) \sqrt {a +b \sinh \left (x \right )}}{105 \sin \left (\frac {\pi }{4}+\frac {i x}{2}\right ) b \sqrt {\frac {a +b \sinh \left (x \right )}{-i b +a}}}-\frac {2 i \left (a^{2}+b^{2}\right ) \left (56 A a b +15 B \,a^{2}-25 b^{2} B \right ) \sqrt {\frac {1}{2}+\frac {i \sinh \left (x \right )}{2}}\, \EllipticF \left (\cos \left (\frac {\pi }{4}+\frac {i x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{i a +b}}\right ) \sqrt {\frac {a +b \sinh \left (x \right )}{-i b +a}}}{105 \sin \left (\frac {\pi }{4}+\frac {i x}{2}\right ) b \sqrt {a +b \sinh \left (x \right )}} \]
command
integrate((a+b*sinh(x))^(5/2)*(A+B*sinh(x)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {8 \, {\left (\sqrt {2} {\left (30 \, B a^{4} + 7 \, A a^{3} b + 115 \, B a^{2} b^{2} + 231 \, A a b^{3} - 75 \, B b^{4}\right )} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} {\left (30 \, B a^{4} + 7 \, A a^{3} b + 115 \, B a^{2} b^{2} + 231 \, A a b^{3} - 75 \, B b^{4}\right )} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} {\left (30 \, B a^{4} + 7 \, A a^{3} b + 115 \, B a^{2} b^{2} + 231 \, A a b^{3} - 75 \, B b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} {\left (30 \, B a^{4} + 7 \, A a^{3} b + 115 \, B a^{2} b^{2} + 231 \, A a b^{3} - 75 \, B b^{4}\right )} \sinh \left (x\right )^{3}\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 ) + 24 \, {\left (\sqrt {2} {\left (15 \, B a^{3} b + 161 \, A a^{2} b^{2} - 145 \, B a b^{3} - 63 \, A b^{4}\right )} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} {\left (15 \, B a^{3} b + 161 \, A a^{2} b^{2} - 145 \, B a b^{3} - 63 \, A b^{4}\right )} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} {\left (15 \, B a^{3} b + 161 \, A a^{2} b^{2} - 145 \, B a b^{3} - 63 \, A b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} {\left (15 \, B a^{3} b + 161 \, A a^{2} b^{2} - 145 \, B a b^{3} - 63 \, A b^{4}\right )} \sinh \left (x\right )^{3}\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 (15 \, B b^{4} \cosh \left (x\right )^{6} + 15 \, B b^{4} \sinh \left (x\right )^{6} + 6 \, {\left (15 \, B a b^{3} + 7 \, A b^{4}\right )} \cosh \left (x\right )^{5} + 6 \, {\left (15 \, B b^{4} \cosh \left (x\right ) + 15 \, B a b^{3} + 7 \, A b^{4}\right )} \sinh \left (x\right )^{5} + 15 \, B b^{4} + {\left (180 \, B a^{2} b^{2} + 308 \, A a b^{3} - 115 \, B b^{4}\right )} \cosh \left (x\right )^{4} + {\left (225 \, B b^{4} \cosh \left (x\right )^{2} + 180 \, B a^{2} b^{2} + 308 \, A a b^{3} - 115 \, B b^{4} + 30 \, {\left (15 \, B a b^{3} + 7 \, A b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} - 8 \, {\left (15 \, B a^{3} b + 161 \, A a^{2} b^{2} - 145 \, B a b^{3} - 63 \, A b^{4}\right )} \cosh \left (x\right )^{3} + 4 \, {\left (75 \, B b^{4} \cosh \left (x\right )^{3} - 30 \, B a^{3} b - 322 \, A a^{2} b^{2} + 290 \, B a b^{3} + 126 \, A b^{4} + 15 \, {\left (15 \, B a b^{3} + 7 \, A b^{4}\right )} \cosh \left (x\right )^{2} + {\left (180 \, B a^{2} b^{2} + 308 \, A a b^{3} - 115 \, B b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (180 \, B a^{2} b^{2} + 308 \, A a b^{3} - 115 \, B b^{4}\right )} \cosh \left (x\right )^{2} + {\left (225 \, B b^{4} \cosh \left (x\right )^{4} + 180 \, B a^{2} b^{2} + 308 \, A a b^{3} - 115 \, B b^{4} + 60 \, {\left (15 \, B a b^{3} + 7 \, A b^{4}\right )} \cosh \left (x\right )^{3} + 6 \, {\left (180 \, B a^{2} b^{2} + 308 \, A a b^{3} - 115 \, B b^{4}\right )} \cosh \left (x\right )^{2} - 24 \, {\left (15 \, B a^{3} b + 161 \, A a^{2} b^{2} - 145 \, B a b^{3} - 63 \, A b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 6 \, {\left (15 \, B a b^{3} + 7 \, A b^{4}\right )} \cosh \left (x\right ) + 2 \, {\left (45 \, B b^{4} \cosh \left (x\right )^{5} - 45 \, B a b^{3} - 21 \, A b^{4} + 15 \, {\left (15 \, B a b^{3} + 7 \, A b^{4}\right )} \cosh \left (x\right )^{4} + 2 \, {\left (180 \, B a^{2} b^{2} + 308 \, A a b^{3} - 115 \, B b^{4}\right )} \cosh \left (x\right )^{3} - 12 \, {\left (15 \, B a^{3} b + 161 \, A a^{2} b^{2} - 145 \, B a b^{3} - 63 \, A b^{4}\right )} \cosh \left (x\right )^{2} + {\left (180 \, B a^{2} b^{2} + 308 \, A a b^{3} - 115 \, B b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}}{1260 \, {\left (b^{2} \cosh \left (x\right )^{3} + 3 \, b^{2} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B b^{2} \sinh \left (x\right )^{3} + A a^{2} + {\left (2 \, B a b + A b^{2}\right )} \sinh \left (x\right )^{2} + {\left (B a^{2} + 2 \, A a b\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}, x\right ) \]