\[ \int \left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)^3 \, dx \]
Optimal antiderivative \[ -\frac {272 c^{2} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3375 a}-\frac {6 c^{2} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{625 a}+\frac {298 c^{2} x \arcsinh \! \left (a x \right )}{75}+\frac {76 a^{2} c^{2} x^{3} \arcsinh \! \left (a x \right )}{225}+\frac {6 a^{4} c^{2} x^{5} \arcsinh \! \left (a x \right )}{125}-\frac {4 c^{2} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arcsinh \! \left (a x \right )^{2}}{15 a}-\frac {3 c^{2} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}} \arcsinh \! \left (a x \right )^{2}}{25 a}+\frac {8 c^{2} x \arcsinh \! \left (a x \right )^{3}}{15}+\frac {4 c^{2} x \left (a^{2} x^{2}+1\right ) \arcsinh \! \left (a x \right )^{3}}{15}+\frac {c^{2} x \left (a^{2} x^{2}+1\right )^{2} \arcsinh \! \left (a x \right )^{3}}{5}-\frac {4144 c^{2} \sqrt {a^{2} x^{2}+1}}{1125 a}-\frac {8 c^{2} \arcsinh \! \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}}{5 a} \]
command
int((a^2*c*x^2+c)^2*arcsinh(a*x)^3,x)
Maple 2022.1 output
\[\int \left (a^{2} c \,x^{2}+c \right )^{2} \arcsinh \left (a x \right )^{3}\, dx\]
Maple 2021.1 output
\[ \frac {c^{2} \left (3375 \arcsinh \left (a x \right )^{3} a^{5} x^{5}-2025 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}+11250 \arcsinh \left (a x \right )^{3} a^{3} x^{3}+810 \arcsinh \left (a x \right ) a^{5} x^{5}-8550 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}-162 \sqrt {a^{2} x^{2}+1}\, x^{4} a^{4}+16875 \arcsinh \left (a x \right )^{3} a x +5700 \arcsinh \left (a x \right ) a^{3} x^{3}-33525 \arcsinh \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-1684 \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}+67050 a x \arcsinh \left (a x \right )-63682 \sqrt {a^{2} x^{2}+1}\right )}{16875 a} \]