Optimal. Leaf size=148 \[ \frac{107 a^2 \sqrt{c-a c x}}{96 x^2}-\frac{149 a^3 \sqrt{c-a c x}}{64 x}+\frac{363}{64} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )-\frac{17 a \sqrt{c-a c x}}{24 x^3}+\frac{\sqrt{c-a c x}}{4 x^4} \]
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Rubi [A] time = 0.305208, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {6167, 6130, 21, 98, 151, 156, 63, 208, 206} \[ \frac{107 a^2 \sqrt{c-a c x}}{96 x^2}-\frac{149 a^3 \sqrt{c-a c x}}{64 x}+\frac{363}{64} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )-\frac{17 a \sqrt{c-a c x}}{24 x^3}+\frac{\sqrt{c-a c x}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 98
Rule 151
Rule 156
Rule 63
Rule 208
Rule 206
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\\ &=-\frac{\int \frac{(c-a c x)^{3/2}}{x^5 (1+a x)} \, dx}{c}\\ &=\frac{\sqrt{c-a c x}}{4 x^4}+\frac{\int \frac{\frac{17 a c^2}{2}-\frac{15}{2} a^2 c^2 x}{x^4 (1+a x) \sqrt{c-a c x}} \, dx}{4 c}\\ &=\frac{\sqrt{c-a c x}}{4 x^4}-\frac{17 a \sqrt{c-a c x}}{24 x^3}-\frac{\int \frac{\frac{107 a^2 c^3}{4}-\frac{85}{4} a^3 c^3 x}{x^3 (1+a x) \sqrt{c-a c x}} \, dx}{12 c^2}\\ &=\frac{\sqrt{c-a c x}}{4 x^4}-\frac{17 a \sqrt{c-a c x}}{24 x^3}+\frac{107 a^2 \sqrt{c-a c x}}{96 x^2}+\frac{\int \frac{\frac{447 a^3 c^4}{8}-\frac{321}{8} a^4 c^4 x}{x^2 (1+a x) \sqrt{c-a c x}} \, dx}{24 c^3}\\ &=\frac{\sqrt{c-a c x}}{4 x^4}-\frac{17 a \sqrt{c-a c x}}{24 x^3}+\frac{107 a^2 \sqrt{c-a c x}}{96 x^2}-\frac{149 a^3 \sqrt{c-a c x}}{64 x}-\frac{\int \frac{\frac{1089 a^4 c^5}{16}-\frac{447}{16} a^5 c^5 x}{x (1+a x) \sqrt{c-a c x}} \, dx}{24 c^4}\\ &=\frac{\sqrt{c-a c x}}{4 x^4}-\frac{17 a \sqrt{c-a c x}}{24 x^3}+\frac{107 a^2 \sqrt{c-a c x}}{96 x^2}-\frac{149 a^3 \sqrt{c-a c x}}{64 x}-\frac{1}{128} \left (363 a^4 c\right ) \int \frac{1}{x \sqrt{c-a c x}} \, dx+\left (4 a^5 c\right ) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx\\ &=\frac{\sqrt{c-a c x}}{4 x^4}-\frac{17 a \sqrt{c-a c x}}{24 x^3}+\frac{107 a^2 \sqrt{c-a c x}}{96 x^2}-\frac{149 a^3 \sqrt{c-a c x}}{64 x}+\frac{1}{64} \left (363 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 )-\left (8 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )\\ &=\frac{\sqrt{c-a c x}}{4 x^4}-\frac{17 a \sqrt{c-a c x}}{24 x^3}+\frac{107 a^2 \sqrt{c-a c x}}{96 x^2}-\frac{149 a^3 \sqrt{c-a c x}}{64 x}+\frac{363}{64} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.111842, size = 109, normalized size = 0.74 \[ \frac{\left (-447 a^3 x^3+214 a^2 x^2-136 a x+48\right ) \sqrt{c-a c x}}{192 x^4}+\frac{363}{64} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 123, normalized size = 0.8 \begin{align*} 2\,{c}^{4}{a}^{4} \left ( -{\frac{1}{{c}^{3}} \left ({\frac{1}{{x}^{4}{a}^{4}{c}^{4}} \left ( -{\frac{149\, \left ( -acx+c \right ) ^{7/2}}{128}}+{\frac{1127\,c \left ( -acx+c \right ) ^{5/2}}{384}}-{\frac{1049\, \left ( -acx+c \right ) ^{3/2}{c}^{2}}{384}}+{\frac{107\,\sqrt{-acx+c}{c}^{3}}{128}} \right ) }-{\frac{363}{128\,\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{-acx+c}}{\sqrt{c}}} \right ) } \right ) }-2\,{\frac{\sqrt{2}}{{c}^{7/2}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-acx+c}\sqrt{2}}{\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 = 1.48473, size = 621, normalized size = 4.2 \begin{align*} \left [\frac{768 \, \sqrt{2} a^{4} \sqrt{c} x^{4} \log \left (\frac{a c x + 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) + 1089 \, 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 (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt{-a c x + c}}{384 \, x^{4}}, \frac{768 \, \sqrt{2} a^{4} \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) - 1089 \, a^{4} \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right ) -{\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt{-a c x + c}}{192 \, x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 44.0773, size = 991, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17103, size = 216, normalized size = 1.46 \begin{align*} \frac{4 \, \sqrt{2} a^{4} c \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} - \frac{363 \, a^{4} c \arctan \left (\frac{\sqrt{-a c x + c}}{\sqrt{-c}}\right )}{64 \, \sqrt{-c}} - \frac{447 \,{\left (a c x - c\right )}^{3} \sqrt{-a c x + c} a^{4} c + 1127 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} a^{4} c^{2} - 1049 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{4} c^{3} + 321 \, \sqrt{-a c x + c} a^{4} c^{4}}{192 \, a^{4} c^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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