\[ \int e^{\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx \]
Optimal antiderivative \[ \frac {16 \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x \sqrt {-a c x +c}}{105 a^{2} \sqrt {1-\frac {1}{a x}}}-\frac {8 \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x^{2} \sqrt {-a c x +c}}{35 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{\frac {3}{2}} x^{3} \sqrt {-a c x +c}}{7 \sqrt {1-\frac {1}{a x}}} \]
command
integrate(1/((a*x-1)/(a*x+1))^(1/2)*x^2*(-a*c*x+c)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {2 \, {\left (\frac {22 \, \sqrt {2} \sqrt {-c}}{a^{2} \mathrm {sgn}\left (c\right )} + \frac {15 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} - 42 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c - 35 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{2}}{a^{2} c^{3} \mathrm {sgn}\left (-a c x - c\right )}\right )}}{105 \, a} \]