\[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 b \,\pi ^{\frac {3}{2}} x}{35 c^{3}}-\frac {b \,\pi ^{\frac {3}{2}} x^{3}}{105 c}-\frac {8 b c \,\pi ^{\frac {3}{2}} x^{5}}{175}-\frac {b \,c^{3} \pi ^{\frac {3}{2}} x^{7}}{49}-\frac {\left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{5 c^{4} \pi }+\frac {\left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {7}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{7 c^{4} \pi ^{2}} \]
command
int(x^3*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x)),x)
Maple 2022.1 output
\[\int x^{3} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )\, dx\]
Maple 2021.1 output
\[ a \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{7 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{35 \pi \,c^{4}}\right )+\frac {b \,\pi ^{\frac {3}{2}} \left (525 \arcsinh \left (c x \right ) c^{8} x^{8}+1365 \arcsinh \left (c x \right ) c^{6} x^{6}-75 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+945 \arcsinh \left (c x \right ) c^{4} x^{4}-168 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}-105 \arcsinh \left (c x \right ) c^{2} x^{2}-35 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-210 \arcsinh \left (c x \right )+210 c x \sqrt {c^{2} x^{2}+1}\right )}{3675 c^{4} \sqrt {c^{2} x^{2}+1}} \]