\[ \int x \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {5 b^{2} d \,x^{2}}{32}+\frac {b^{2} c^{2} d \,x^{4}}{32}-\frac {b d x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{8 c}-\frac {3 d \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{32 c^{2}}+\frac {d \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{4 c^{2}}-\frac {3 b d x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{16 c} \]
command
int(x*(c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x \left (c^{2} d \,x^{2}+d \right ) \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d \,a^{2} \left (\frac {1}{4} c^{4} x^{4}+\frac {1}{2} c^{2} x^{2}\right )+d \,b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{2}}{4}-\frac {\arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}-\frac {3 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{16}-\frac {3 \arcsinh \left (c x \right )^{2}}{32}+\frac {\left (c^{2} x^{2}+1\right )^{2}}{32}+\frac {3 c^{2} x^{2}}{32}+\frac {3}{32}\right )+2 d a b \left (\frac {\arcsinh \left (c x \right ) c^{4} x^{4}}{4}+\frac {\arcsinh \left (c x \right ) c^{2} x^{2}}{2}-\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{16}-\frac {5 c x \sqrt {c^{2} x^{2}+1}}{32}+\frac {5 \arcsinh \left (c x \right )}{32}\right )}{c^{2}} \]