\[ \int \left (\pi +c^2 \pi x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {65 b^{2} \pi ^{\frac {5}{2}} x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{1728}+\frac {b^{2} \pi ^{\frac {5}{2}} x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{108}-\frac {115 b^{2} \pi ^{\frac {5}{2}} \arcsinh \! \left (c x \right )}{1152 c}-\frac {5 b c \,\pi ^{\frac {5}{2}} x^{2} \left (a +b \arcsinh \! \left (c x \right )\right )}{16}-\frac {5 b \,\pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )}{48 c}-\frac {b \,\pi ^{\frac {5}{2}} \left (c^{2} x^{2}+1\right )^{3} \left (a +b \arcsinh \! \left (c x \right )\right )}{18 c}+\frac {5 \pi x \left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{24}+\frac {x \left (c^{2} \pi \,x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{6}+\frac {5 \pi ^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )^{3}}{48 b c}+\frac {245 b^{2} \pi ^{\frac {5}{2}} x \sqrt {c^{2} x^{2}+1}}{1152}+\frac {5 \pi ^{2} x \left (a +b \arcsinh \! \left (c x \right )\right )^{2} \sqrt {c^{2} \pi \,x^{2}+\pi }}{16} \]
command
int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {a^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{6}+\frac {5 a^{2} \pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{24}+\frac {5 a^{2} \pi ^{2} x \sqrt {\pi \,c^{2} x^{2}+\pi }}{16}+\frac {5 a^{2} \pi ^{3} \ln \left (\frac {\pi x \,c^{2}}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{16 \sqrt {\pi \,c^{2}}}+\frac {b^{2} \pi ^{\frac {5}{2}} c^{4} \sqrt {c^{2} x^{2}+1}\, \arcsinh \left (c x \right )^{2} x^{5}}{6}-\frac {b^{2} \pi ^{\frac {5}{2}} c^{5} \arcsinh \left (c x \right ) x^{6}}{18}+\frac {b^{2} \pi ^{\frac {5}{2}} c^{4} x^{5} \sqrt {c^{2} x^{2}+1}}{108}+\frac {13 b^{2} \pi ^{\frac {5}{2}} c^{2} \sqrt {c^{2} x^{2}+1}\, \arcsinh \left (c x \right )^{2} x^{3}}{24}-\frac {13 b^{2} \pi ^{\frac {5}{2}} c^{3} \arcsinh \left (c x \right ) x^{4}}{48}+\frac {97 b^{2} \pi ^{\frac {5}{2}} c^{2} x^{3} \sqrt {c^{2} x^{2}+1}}{1728}+\frac {11 b^{2} \pi ^{\frac {5}{2}} \arcsinh \left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x}{16}-\frac {11 b^{2} \pi ^{\frac {5}{2}} c \arcsinh \left (c x \right ) x^{2}}{16}+\frac {299 b^{2} \pi ^{\frac {5}{2}} x \sqrt {c^{2} x^{2}+1}}{1152}+\frac {5 b^{2} \pi ^{\frac {5}{2}} \arcsinh \left (c x \right )^{3}}{48 c}-\frac {299 b^{2} \pi ^{\frac {5}{2}} \arcsinh \left (c x \right )}{1152 c}+\frac {a b \,\pi ^{\frac {5}{2}} c^{4} \sqrt {c^{2} x^{2}+1}\, \arcsinh \left (c x \right ) x^{5}}{3}-\frac {a b \,\pi ^{\frac {5}{2}} c^{5} x^{6}}{18}+\frac {13 a b \,\pi ^{\frac {5}{2}} c^{2} \sqrt {c^{2} x^{2}+1}\, \arcsinh \left (c x \right ) x^{3}}{12}-\frac {13 a b \,\pi ^{\frac {5}{2}} c^{3} x^{4}}{48}+\frac {11 a b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{8}-\frac {11 a b \,\pi ^{\frac {5}{2}} c \,x^{2}}{16}+\frac {5 a b \,\pi ^{\frac {5}{2}} \arcsinh \left (c x \right )^{2}}{16 c}-\frac {17 a b \,\pi ^{\frac {5}{2}}}{36 c} \]