\[ \int \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {4322 b^{2} d^{3} x}{3675}+\frac {1514 b^{2} c^{2} d^{3} x^{3}}{11025}+\frac {234 b^{2} c^{4} d^{3} x^{5}}{6125}+\frac {2 b^{2} c^{6} d^{3} x^{7}}{343}-\frac {16 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{105 c}-\frac {12 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{175 c}-\frac {2 b \,d^{3} \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{49 c}+\frac {16 d^{3} x \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{35}+\frac {8 d^{3} x \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{35}+\frac {6 d^{3} x \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{35}+\frac {d^{3} x \left (c^{2} x^{2}+1\right )^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{7}-\frac {32 b \,d^{3} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{35 c} \]
command
int((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d^{3} a^{2} \left (\frac {1}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}+c x \right )+d^{3} b^{2} \left (\frac {16 \arcsinh \left (c x \right )^{2} c x}{35}+\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{7}+\frac {6 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{35}+\frac {8 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{35}-\frac {32 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{35}+\frac {413312 c x}{385875}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{49}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {888 c x \left (c^{2} x^{2}+1\right )^{2}}{42875}+\frac {30256 c x \left (c^{2} x^{2}+1\right )}{385875}-\frac {12 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{175}-\frac {16 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{105}\right )+2 d^{3} a b \left (\frac {\arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\arcsinh \left (c x \right ) c^{3} x^{3}+\arcsinh \left (c x \right ) c x -\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {117 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {757 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c} \]