Optimal. Leaf size=195 \[ \frac{184 (c-a c x)^{3/2}}{5 a^3 x^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^3 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{16 (c-a c x)^{3/2}}{a^2 x \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{8 (c-a c x)^{3/2}}{5 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.191599, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6176, 6181, 94, 89, 78, 37} \[ \frac{184 (c-a c x)^{3/2}}{5 a^3 x^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^3 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{16 (c-a c x)^{3/2}}{a^2 x \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{8 (c-a c x)^{3/2}}{5 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 94
Rule 89
Rule 78
Rule 37
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\frac{(c-a c x)^{3/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac{1}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3}{x^{7/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{3/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (12 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\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 )}{5 a \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{8 (c-a c x)^{3/2}}{5 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{2 \left (a-\frac{1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (8 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\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 )}{5 a \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{8 (c-a c x)^{3/2}}{5 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{16 (c-a c x)^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (92 \left (\frac{1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2}}\\ &=-\frac{8 (c-a c x)^{3/2}}{5 a \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{184 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{16 (c-a c x)^{3/2}}{a^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0322588, size = 57, normalized size = 0.29 \[ -\frac{2 c \left (a^3 x^3-7 a^2 x^2+43 a x+91\right ) \sqrt{c-a c x}}{5 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 63, normalized size = 0.3 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ({x}^{3}{a}^{3}-7\,{a}^{2}{x}^{2}+43\,ax+91 \right ) }{5\,a \left ( ax-1 \right ) ^{3}} \left ( -acx+c \right ) ^{{\frac{3}{2}}} \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 = 1.07883, size = 126, normalized size = 0.65 \begin{align*} -\frac{2 \,{\left (a^{4} \sqrt{-c} c x^{4} - 6 \, a^{3} \sqrt{-c} c x^{3} + 36 \, a^{2} \sqrt{-c} c x^{2} + 134 \, a \sqrt{-c} c x + 91 \, \sqrt{-c} c\right )}{\left (a x - 1\right )}^{2}}{5 \,{\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 = 1.56261, size = 142, normalized size = 0.73 \begin{align*} -\frac{2 \,{\left (a^{3} c x^{3} - 7 \, a^{2} c x^{2} + 43 \, a c x + 91 \, c\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{5 \,{\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 = 1.20292, size = 104, normalized size = 0.53 \begin{align*} \frac{2 \,{\left ({\left (a c x + c\right )}^{2} \sqrt{-a c x - c} + 10 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c + 60 \, \sqrt{-a c x - c} c^{2} - \frac{40 \, c^{3}}{\sqrt{-a c x - c}}\right )}{\left | c \right |}}{5 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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