Optimal. Leaf size=182 \[ \frac{316 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{158 \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{32 x \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.216173, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6176, 6181, 89, 78, 45, 37} \[ \frac{316 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{158 \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{32 x \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 89
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} 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}} x^{3/2} \, 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{\left (1-\frac{x}{a}\right )^2}{x^{7/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 x^2 \sqrt{c-a c x}}{5 \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{8}{a}+\frac{5 x}{2 a^2}}{x^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{5 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{32 x \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{2 x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (79 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{15 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{158 \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{32 x \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{2 x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (158 \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 )}{15 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{158 \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{316 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{32 x \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{2 x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.0302175, size = 57, normalized size = 0.31 \[ \frac{2 \left (3 a^3 x^3-16 a^2 x^2+79 a x+158\right ) \sqrt{c-a c x}}{15 a^3 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 64, normalized size = 0.4 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 3\,{x}^{3}{a}^{3}-16\,{a}^{2}{x}^{2}+79\,ax+158 \right ) }{15\,{a}^{2} \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 = 1.11408, size = 123, normalized size = 0.68 \begin{align*} \frac{2 \,{\left (3 \, a^{4} \sqrt{-c} x^{4} - 13 \, a^{3} \sqrt{-c} x^{3} + 63 \, a^{2} \sqrt{-c} x^{2} + 237 \, a \sqrt{-c} x + 158 \, \sqrt{-c}\right )}{\left (a x - 1\right )}^{2}}{15 \,{\left (a^{4} x^{2} - 2 \, a^{3} x + a^{2}\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 = 1.39988, size = 139, normalized size = 0.76 \begin{align*} \frac{2 \,{\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 79 \, a x + 158\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{3} x - a^{2}\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 = 1.19304, size = 113, normalized size = 0.62 \begin{align*} -\frac{2 \,{\left (3 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c}{\left | c \right |} + 25 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c{\left | c \right |} + 120 \, \sqrt{-a c x - c} c^{2}{\left | c \right |} - \frac{60 \, c^{3}{\left | c \right |}}{\sqrt{-a c x - c}}\right )}}{15 \, a^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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