\[ \int x^3 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ -\frac {b^{2} d \,x^{2}}{24 c^{2}}+\frac {b^{2} d \,x^{4}}{72}+\frac {b^{2} c^{2} d \,x^{6}}{108}-\frac {d \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{24 c^{4}}+\frac {d \,x^{4} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{12}+\frac {d \,x^{4} \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{6}+\frac {b d x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{12 c^{3}}-\frac {b d \,x^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{18 c}-\frac {b c d \,x^{5} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{18} \]
command
int(x^3*(c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x^{3} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d \,a^{2} \left (\frac {1}{6} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d \,b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{2}}{6}-\frac {\arcsinh \left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{12}-\frac {\arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{18}+\frac {\arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{18}+\frac {\arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{12}+\frac {\arcsinh \left (c x \right )^{2}}{24}+\frac {\left (c^{2} x^{2}+1\right )^{3}}{108}-\frac {\left (c^{2} x^{2}+1\right )^{2}}{72}-\frac {c^{2} x^{2}}{24}-\frac {1}{24}\right )+2 d a b \left (\frac {\arcsinh \left (c x \right ) c^{6} x^{6}}{6}+\frac {\arcsinh \left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{36}-\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}+\frac {c x \sqrt {c^{2} x^{2}+1}}{24}-\frac {\arcsinh \left (c x \right )}{24}\right )}{c^{4}} \]