\[ \int x^3 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ -\frac {73 b^{2} d^{2} x^{2}}{3072 c^{2}}+\frac {73 b^{2} d^{2} x^{4}}{9216}+\frac {43 b^{2} c^{2} d^{2} x^{6}}{3456}+\frac {b^{2} c^{4} d^{2} x^{8}}{256}-\frac {b c \,d^{2} x^{5} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{32}-\frac {73 d^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{3072 c^{4}}+\frac {d^{2} x^{4} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{24}+\frac {d^{2} x^{4} \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{12}+\frac {d^{2} x^{4} \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{8}+\frac {73 b \,d^{2} x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{1536 c^{3}}-\frac {73 b \,d^{2} x^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{2304 c}-\frac {25 b c \,d^{2} x^{5} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{576} \]
command
int(x^3*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d^{2} a^{2} \left (\frac {1}{8} c^{8} x^{8}+\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{3}}{8}-\frac {\arcsinh \left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{3}}{24}-\frac {\arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}+\frac {11 \arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{576}+\frac {55 \arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{2304}+\frac {55 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{1536}+\frac {55 \arcsinh \left (c x \right )^{2}}{3072}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}-\frac {11 \left (c^{2} x^{2}+1\right )^{3}}{3456}-\frac {55 \left (c^{2} x^{2}+1\right )^{2}}{9216}-\frac {55 c^{2} x^{2}}{3072}-\frac {55}{3072}\right )+2 d^{2} a b \left (\frac {\arcsinh \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsinh \left (c x \right ) c^{6} x^{6}}{3}+\frac {\arcsinh \left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{1152}-\frac {73 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {c^{2} x^{2}+1}}{3072}-\frac {73 \arcsinh \left (c x \right )}{3072}\right )}{c^{4}} \]