\[ \int x^2 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ -\frac {52 b^{2} d x}{225 c^{2}}+\frac {26 b^{2} d \,x^{3}}{675}+\frac {2 b^{2} c^{2} d \,x^{5}}{125}+\frac {2 b d \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{15 c^{3}}-\frac {2 b d \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{25 c^{3}}+\frac {2 d \,x^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{15}+\frac {d \,x^{3} \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{5}+\frac {8 b d \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{45 c^{3}}-\frac {4 b d \,x^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{45 c} \]
command
int(x^2*(c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x^{2} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d \,a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d \,b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}-\frac {2 \arcsinh \left (c x \right )^{2} c x}{15}-\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}-\frac {856 c x}{3375}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {22 c x \left (c^{2} x^{2}+1\right )}{3375}+\frac {4 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 d a b \left (\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\frac {\arcsinh \left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {13 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c^{3}} \]