\[ \int x^3 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ -\frac {79 b^{2} d^{3} x^{2}}{5120 c^{2}}+\frac {79 b^{2} d^{3} x^{4}}{15360}+\frac {401 b^{2} c^{2} d^{3} x^{6}}{28800}+\frac {57 b^{2} c^{4} d^{3} x^{8}}{6400}+\frac {b^{2} c^{6} d^{3} x^{10}}{500}-\frac {b c \,d^{3} x^{5} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{32}-\frac {b c \,d^{3} x^{5} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{50}-\frac {79 d^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{5120 c^{4}}+\frac {d^{3} x^{4} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{40}+\frac {d^{3} x^{4} \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{20}+\frac {3 d^{3} x^{4} \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{40}+\frac {d^{3} x^{4} \left (c^{2} x^{2}+1\right )^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{10}+\frac {79 b \,d^{3} x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{2560 c^{3}}-\frac {79 b \,d^{3} x^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{3840 c}-\frac {31 b c \,d^{3} x^{5} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{960} \]
command
int(x^3*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x^{3} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d^{3} a^{2} \left (\frac {1}{10} c^{10} x^{10}+\frac {3}{8} c^{8} x^{8}+\frac {1}{2} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{3} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{4}}{10}-\frac {\arcsinh \left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{4}}{40}-\frac {\arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{50}+\frac {7 \arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{800}+\frac {49 \arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{4800}+\frac {49 \arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3840}+\frac {49 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{2560}+\frac {49 \arcsinh \left (c x \right )^{2}}{5120}+\frac {\left (c^{2} x^{2}+1\right )^{5}}{500}-\frac {7 \left (c^{2} x^{2}+1\right )^{4}}{6400}-\frac {49 \left (c^{2} x^{2}+1\right )^{3}}{28800}-\frac {49 \left (c^{2} x^{2}+1\right )^{2}}{15360}-\frac {49 c^{2} x^{2}}{5120}-\frac {49}{5120}\right )+2 d^{3} a b \left (\frac {\arcsinh \left (c x \right ) c^{10} x^{10}}{10}+\frac {3 \arcsinh \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsinh \left (c x \right ) c^{6} x^{6}}{2}+\frac {\arcsinh \left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{9} x^{9} \sqrt {c^{2} x^{2}+1}}{100}-\frac {57 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{1600}-\frac {401 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{9600}-\frac {79 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{7680}+\frac {79 c x \sqrt {c^{2} x^{2}+1}}{5120}-\frac {79 \arcsinh \left (c x \right )}{5120}\right )}{c^{4}} \]