\[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (A b -B a \right ) \cosh \left (x \right )}{3 \left (a^{2}+b^{2}\right ) \left (a +b \sinh \left (x \right )\right )^{\frac {3}{2}}}-\frac {2 \left (4 A a b -B \,a^{2}+3 b^{2} B \right ) \cosh \left (x \right )}{3 \left (a^{2}+b^{2}\right )^{2} \sqrt {a +b \sinh \left (x \right )}}+\frac {2 i \left (4 A a b -B \,a^{2}+3 b^{2} B \right ) \sqrt {\frac {1}{2}+\frac {i \sinh \left (x \right )}{2}}\, \EllipticE \left (\cos \left (\frac {\pi }{4}+\frac {i x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{i a +b}}\right ) \sqrt {a +b \sinh \left (x \right )}}{3 \sin \left (\frac {\pi }{4}+\frac {i x}{2}\right ) b \left (a^{2}+b^{2}\right )^{2} \sqrt {\frac {a +b \sinh \left (x \right )}{-i b +a}}}-\frac {2 i \left (A b -B a \right ) \sqrt {\frac {1}{2}+\frac {i \sinh \left (x \right )}{2}}\, \EllipticF \left (\cos \left (\frac {\pi }{4}+\frac {i x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{i a +b}}\right ) \sqrt {\frac {a +b \sinh \left (x \right )}{-i b +a}}}{3 \sin \left (\frac {\pi }{4}+\frac {i x}{2}\right ) b \left (a^{2}+b^{2}\right ) \sqrt {a +b \sinh \left (x \right )}} \]
command
integrate((A+B*sinh(x))/(a+b*sinh(x))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {{\left (B \sinh \left (x\right ) + A\right )} \sqrt {b \sinh \left (x\right ) + a}}{b^{3} \sinh \left (x\right )^{3} + 3 \, a b^{2} \sinh \left (x\right )^{2} + 3 \, a^{2} b \sinh \left (x\right ) + a^{3}}, x\right ) \]