Optimal. Leaf size=137 \[ -\frac{46 \sqrt{c-a c x}}{3 a^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{20 \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.155477, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6176, 6181, 89, 78, 37} \[ -\frac{46 \sqrt{c-a c x}}{3 a^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{20 \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 89
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} \sqrt{x} \, 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{\left (1-\frac{x}{a}\right )^2}{x^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{5}{a}+\frac{3 x}{2 a^2}}{x^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{20 \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (23 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{20 \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{46 \sqrt{c-a c x}}{3 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}+\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0268875, size = 48, normalized size = 0.35 \[ \frac{2 \left (a^2 x^2-10 a x-23\right ) \sqrt{c-a c x}}{3 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 55, normalized size = 0.4 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ({a}^{2}{x}^{2}-10\,ax-23 \right ) }{3\,a \left ( ax-1 \right ) ^{2}}\sqrt{-acx+c} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0863, size = 101, normalized size = 0.74 \begin{align*} \frac{2 \,{\left (a^{3} \sqrt{-c} x^{3} - 9 \, a^{2} \sqrt{-c} x^{2} - 33 \, a \sqrt{-c} x - 23 \, \sqrt{-c}\right )}{\left (a x - 1\right )}^{2}}{3 \,{\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )}{\left (a x + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58946, size = 113, normalized size = 0.82 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} - 10 \, a x - 23\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{2} x - a\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 = 1.19042, size = 70, normalized size = 0.51 \begin{align*} \frac{2 \,{\left ({\left (-a c x - c\right )}^{\frac{3}{2}} + 12 \, \sqrt{-a c x - c} c - \frac{12 \, c^{2}}{\sqrt{-a c x - c}}\right )}{\left | c \right |}}{3 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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