Optimal. Leaf size=142 \[ -\frac{2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}}{7 a c}+\frac{6 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}}{35 a^2 c}+\frac{38 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}}{105 a^2}+\frac{152 c x \sqrt{1-\frac{1}{a^2 x^2}}}{105 a^2 \sqrt{c-a c x}} \]
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Rubi [A] time = 0.236783, antiderivative size = 185, normalized size of antiderivative = 1.3, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6176, 6181, 78, 45, 37} \[ \frac{104 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{208 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}-\frac{26 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Rule 6176
Rule 6181
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{-\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^{5/2} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{x}{a}}{x^{9/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{\left (13 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{7/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{7 a \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{26 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}-\frac{\left (52 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{5/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{35 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{104 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{26 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{\left (104 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{105 a^3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{208 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{105 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{104 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{26 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0344716, size = 67, normalized size = 0.47 \[ \frac{2 \sqrt{\frac{1}{a x}+1} \left (15 a^3 x^3-39 a^2 x^2+52 a x-104\right ) \sqrt{c-a c x}}{105 a^3 \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 64, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 15\,{x}^{3}{a}^{3}-39\,{a}^{2}{x}^{2}+52\,ax-104 \right ) }{105\,{a}^{3} \left ( ax-1 \right ) }\sqrt{-acx+c}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10097, size = 112, normalized size = 0.79 \begin{align*} \frac{2 \,{\left (15 \, a^{4} \sqrt{-c} x^{4} - 24 \, a^{3} \sqrt{-c} x^{3} + 13 \, a^{2} \sqrt{-c} x^{2} - 52 \, a \sqrt{-c} x - 104 \, \sqrt{-c}\right )}{\left (a x - 1\right )}}{105 \,{\left (a^{4} x - a^{3}\right )} \sqrt{a x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63865, size = 159, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (15 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 13 \, a^{2} x^{2} - 52 \, a x - 104\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{105 \,{\left (a^{4} x - a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17414, size = 150, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (\frac{76 \, \sqrt{2} \sqrt{-c} c}{a^{3}} + \frac{15 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} - 84 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c - 175 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{2} - 210 \, \sqrt{-a c x - c} c^{3}}{a^{3} c^{2}}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{105 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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