\[ \int x^2 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ -\frac {1636 b^{2} d^{2} x}{11025 c^{2}}+\frac {818 b^{2} d^{2} x^{3}}{33075}+\frac {136 b^{2} c^{2} d^{2} x^{5}}{6125}+\frac {2 b^{2} c^{4} d^{2} x^{7}}{343}+\frac {8 b \,d^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{105 c^{3}}+\frac {2 b \,d^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{175 c^{3}}-\frac {2 b \,d^{2} \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{49 c^{3}}+\frac {8 d^{2} x^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{105}+\frac {4 d^{2} x^{3} \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{35}+\frac {d^{2} x^{3} \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{7}+\frac {32 b \,d^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{315 c^{3}}-\frac {16 b \,d^{2} x^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{315 c} \]
command
int(x^2*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x^{2} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d^{2} a^{2} \left (\frac {1}{7} c^{7} x^{7}+\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{7}-\frac {8 \arcsinh \left (c x \right )^{2} c x}{105}-\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{35}-\frac {4 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{105}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{49}-\frac {181456 c x}{1157625}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {202 c x \left (c^{2} x^{2}+1\right )^{2}}{42875}-\frac {2528 c x \left (c^{2} x^{2}+1\right )}{1157625}+\frac {16 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{105}+\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{175}+\frac {8 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{315}\right )+2 d^{2} a b \left (\frac {\arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {2 \arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\frac {\arcsinh \left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {68 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {409 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {c^{2} x^{2}+1}}{11025}\right )}{c^{3}} \]