\[ \int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx \]
Optimal antiderivative \[ \frac {6 x \left (-a c x +c \right )^{\frac {3}{2}} \sqrt {1-\frac {1}{a^{2} x^{2}}}}{35 a^{2} c}-\frac {2 x^{2} \left (-a c x +c \right )^{\frac {3}{2}} \sqrt {1-\frac {1}{a^{2} x^{2}}}}{7 a c}+\frac {152 c x \sqrt {1-\frac {1}{a^{2} x^{2}}}}{105 a^{2} \sqrt {-a c x +c}}+\frac {38 x \sqrt {1-\frac {1}{a^{2} x^{2}}}\, \sqrt {-a c x +c}}{105 a^{2}} \]
command
integrate(x**2*(-a*c*x+c)**(1/2)*((a*x-1)/(a*x+1))**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {304 c \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}}{105 a^{3} \sqrt {- a c x + c}} - \frac {76 \sqrt {- a c x + c} \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}}{105 a^{3}} - \frac {62 \left (- a c x + c\right )^{\frac {3}{2}} \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}}{105 a^{3} c} + \frac {24 \left (- a c x + c\right )^{\frac {5}{2}} \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}}{35 a^{3} c^{2}} - \frac {2 \left (- a c x + c\right )^{\frac {7}{2}} \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}}{7 a^{3} c^{3}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________