Optimal. Leaf size=110 \[ \frac{25 a^2 \sqrt{c-a c x}}{32 x^2}+\frac{75 a^3 \sqrt{c-a c x}}{64 x}+\frac{75}{64} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+\frac{5 a \sqrt{c-a c x}}{8 x^3}+\frac{\sqrt{c-a c x}}{4 x^4} \]
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Rubi [A] time = 0.234181, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{25 a^2 \sqrt{c-a c x}}{32 x^2}+\frac{75 a^3 \sqrt{c-a c x}}{64 x}+\frac{75}{64} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )+\frac{5 a \sqrt{c-a c x}}{8 x^3}+\frac{\sqrt{c-a c x}}{4 x^4} \]
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^5} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^5} \, dx\\ &=-\int \frac{(1+a x) \sqrt{c-a c x}}{x^5 (1-a x)} \, dx\\ &=-\left (c \int \frac{1+a x}{x^5 \sqrt{c-a c x}} \, dx\right )\\ &=\frac{\sqrt{c-a c x}}{4 x^4}-\frac{1}{8} (15 a c) \int \frac{1}{x^4 \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{4 x^4}+\frac{5 a \sqrt{c-a c x}}{8 x^3}-\frac{1}{16} \left (25 a^2 c\right ) \int \frac{1}{x^3 \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{4 x^4}+\frac{5 a \sqrt{c-a c x}}{8 x^3}+\frac{25 a^2 \sqrt{c-a c x}}{32 x^2}-\frac{1}{64} \left (75 a^3 c\right ) \int \frac{1}{x^2 \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{4 x^4}+\frac{5 a \sqrt{c-a c x}}{8 x^3}+\frac{25 a^2 \sqrt{c-a c x}}{32 x^2}+\frac{75 a^3 \sqrt{c-a c x}}{64 x}-\frac{1}{128} \left (75 a^4 c\right ) \int \frac{1}{x \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{4 x^4}+\frac{5 a \sqrt{c-a c x}}{8 x^3}+\frac{25 a^2 \sqrt{c-a c x}}{32 x^2}+\frac{75 a^3 \sqrt{c-a c x}}{64 x}+\frac{1}{64} \left (75 a^3\right ) \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}}{4 x^4}+\frac{5 a \sqrt{c-a c x}}{8 x^3}+\frac{25 a^2 \sqrt{c-a c x}}{32 x^2}+\frac{75 a^3 \sqrt{c-a c x}}{64 x}+\frac{75}{64} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0685087, size = 71, normalized size = 0.65 \[ \frac{\left (75 a^3 x^3+50 a^2 x^2+40 a x+16\right ) \sqrt{c-a c x}}{64 x^4}+\frac{75}{64} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 93, normalized size = 0.9 \begin{align*} 2\,{c}^{4}{a}^{4} \left ({\frac{1}{{x}^{4}{a}^{4}{c}^{4}} \left ( -{\frac{75\, \left ( -acx+c \right ) ^{7/2}}{128\,{c}^{3}}}+{\frac{275\, \left ( -acx+c \right ) ^{5/2}}{128\,{c}^{2}}}-{\frac{365\, \left ( -acx+c \right ) ^{3/2}}{128\,c}}+{\frac{181\,\sqrt{-acx+c}}{128}} \right ) }+{\frac{75}{128\,{c}^{7/2}}{\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 = 1.62456, size = 371, normalized size = 3.37 \begin{align*} \left [\frac{75 \, a^{4} \sqrt{c} x^{4} \log \left (\frac{a c x - 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right ) + 2 \,{\left (75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right )} \sqrt{-a c x + c}}{128 \, x^{4}}, -\frac{75 \, a^{4} \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right ) -{\left (75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right )} \sqrt{-a c x + c}}{64 \, x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 27.5307, size = 639, normalized size = 5.81 \begin{align*} \frac{558 a^{4} c^{8} \sqrt{- a c x + c}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} - \frac{1022 a^{4} c^{7} \left (- a c x + c\right )^{\frac{3}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} + \frac{770 a^{4} c^{6} \left (- a c x + c\right )^{\frac{5}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} - \frac{66 a^{4} c^{6} \sqrt{- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac{210 a^{4} c^{5} \left (- a c x + c\right )^{\frac{7}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} + \frac{80 a^{4} c^{5} \left (- a c x + c\right )^{\frac{3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac{35 a^{4} c^{5} \sqrt{\frac{1}{c^{9}}} \log{\left (- c^{5} \sqrt{\frac{1}{c^{9}}} + \sqrt{- a c x + c} \right )}}{128} + \frac{35 a^{4} c^{5} \sqrt{\frac{1}{c^{9}}} \log{\left (c^{5} \sqrt{\frac{1}{c^{9}}} + \sqrt{- a c x + c} \right )}}{128} - \frac{30 a^{4} c^{4} \left (- a c x + c\right )^{\frac{5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac{5 a^{4} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left (- c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right )}}{16} + \frac{5 a^{4} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left (c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17964, size = 170, normalized size = 1.55 \begin{align*} -\frac{75 \, a^{4} c \arctan \left (\frac{\sqrt{-a c x + c}}{\sqrt{-c}}\right )}{64 \, \sqrt{-c}} + \frac{75 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} a^{4} c + 275 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} a^{4} c^{2} - 365 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{4} c^{3} + 181 \, \sqrt{-a c x + c} a^{4} c^{4}}{64 \, a^{4} c^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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