Optimal. Leaf size=68 \[ \frac{7}{4} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+\frac{\sqrt{c-a c x}}{2 x^2}+\frac{7 a \sqrt{c-a c x}}{4 x} \]
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Rubi [A] time = 0.210414, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6167, 6130, 21, 78, 51, 63, 208} \[ \frac{7}{4} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+\frac{\sqrt{c-a c x}}{2 x^2}+\frac{7 a \sqrt{c-a c x}}{4 x} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx\\ &=-\int \frac{(1+a x) \sqrt{c-a c x}}{x^3 (1-a x)} \, dx\\ &=-\left (c \int \frac{1+a x}{x^3 \sqrt{c-a c x}} \, dx\right )\\ &=\frac{\sqrt{c-a c x}}{2 x^2}-\frac{1}{4} (7 a c) \int \frac{1}{x^2 \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{2 x^2}+\frac{7 a \sqrt{c-a c x}}{4 x}-\frac{1}{8} \left (7 a^2 c\right ) \int \frac{1}{x \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{2 x^2}+\frac{7 a \sqrt{c-a c x}}{4 x}+\frac{1}{4} (7 a) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a c}} \, dx,x,\sqrt{c-a c x}\right )\\ &=\frac{\sqrt{c-a c x}}{2 x^2}+\frac{7 a \sqrt{c-a c x}}{4 x}+\frac{7}{4} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0472039, size = 55, normalized size = 0.81 \[ \frac{7}{4} a^2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+\frac{(7 a x+2) \sqrt{c-a c x}}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 65, normalized size = 1. \begin{align*} 2\,{a}^{2}{c}^{2} \left ({\frac{1}{{a}^{2}{x}^{2}{c}^{2}} \left ( -{\frac{7\, \left ( -acx+c \right ) ^{3/2}}{8\,c}}+{\frac{9\,\sqrt{-acx+c}}{8}} \right ) }+{\frac{7}{8\,{c}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{-acx+c}}{\sqrt{c}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59502, size = 289, normalized size = 4.25 \begin{align*} \left [\frac{7 \, a^{2} \sqrt{c} x^{2} \log \left (\frac{a c x - 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right ) + 2 \, \sqrt{-a c x + c}{\left (7 \, a x + 2\right )}}{8 \, x^{2}}, -\frac{7 \, a^{2} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right ) - \sqrt{-a c x + c}{\left (7 \, a x + 2\right )}}{4 \, x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 23.2499, size = 270, normalized size = 3.97 \begin{align*} \frac{10 a^{2} c^{4} \sqrt{- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac{6 a^{2} c^{3} \left (- a c x + c\right )^{\frac{3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac{3 a^{2} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left (- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right )}}{8} + \frac{3 a^{2} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left (c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right )}}{8} - \frac{a^{2} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right )}}{2} + \frac{a^{2} c^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{- a c x + c} \right )}}{2} + \frac{a \sqrt{- a c x + c}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1418, size = 97, normalized size = 1.43 \begin{align*} -\frac{7 \, a^{2} c \arctan \left (\frac{\sqrt{-a c x + c}}{\sqrt{-c}}\right )}{4 \, \sqrt{-c}} - \frac{7 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{2} c - 9 \, \sqrt{-a c x + c} a^{2} c^{2}}{4 \, a^{2} c^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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