Optimal. Leaf size=221 \[ \frac{x \left (a-\frac{1}{x}\right )^3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{3 \left (a-\frac{1}{x}\right )^2 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{\left (80 a-\frac{7}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{9 \left (c-\frac{c}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{7/2}} \]
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Rubi [A] time = 0.144436, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6182, 6179, 98, 153, 147, 63, 208} \[ \frac{x \left (a-\frac{1}{x}\right )^3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{3 \left (a-\frac{1}{x}\right )^2 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{\left (80 a-\frac{7}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{9 \left (c-\frac{c}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 98
Rule 153
Rule 147
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{7/2} \, dx &=\frac{\left (c-\frac{c}{a x}\right )^{7/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{7/2} \, dx}{\left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{\left (c-\frac{c}{a x}\right )^{7/2} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^4}{x^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{7/2}}\\ &=\frac{\left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (c-\frac{c}{a x}\right )^{7/2} \operatorname{Subst}\left (\int \frac{\left (\frac{9}{2 a}+\frac{3 x}{2 a^2}\right ) \left (1-\frac{x}{a}\right )^2}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{7/2}}\\ &=\frac{3 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (2 a \left (c-\frac{c}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{45}{4 a^2}-\frac{21 x}{4 a^3}\right ) \left (1-\frac{x}{a}\right )}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5 \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{\left (80 a-\frac{7}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{3 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (9 \left (c-\frac{c}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{\left (80 a-\frac{7}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{3 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (9 \left (c-\frac{c}{a x}\right )^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\left (1-\frac{1}{a x}\right )^{7/2}}\\ &=-\frac{\left (80 a-\frac{7}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{3 \left (a-\frac{1}{x}\right )^2 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{9 \left (c-\frac{c}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \left (1-\frac{1}{a x}\right )^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.102816, size = 101, normalized size = 0.46 \[ \frac{c^3 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (5 a^3 x^3-92 a^2 x^2+16 a x-2\right )-45 a^2 x^2 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{5 a^3 x^2 \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 161, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax+1 \right ){c}^{3}}{10\,{x}^{2} \left ( ax-1 \right ) }\sqrt{{\frac{ax-1}{ax+1}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 10\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}-184\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}-45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{3}{a}^{3}+32\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-4\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ){a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.63215, size = 859, normalized size = 3.89 \begin{align*} \left [\frac{45 \,{\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (5 \, a^{4} c^{3} x^{4} - 87 \, a^{3} c^{3} x^{3} - 76 \, a^{2} c^{3} x^{2} + 14 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{20 \,{\left (a^{4} x^{3} - a^{3} x^{2}\right )}}, \frac{45 \,{\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (5 \, a^{4} c^{3} x^{4} - 87 \, a^{3} c^{3} x^{3} - 76 \, a^{2} c^{3} x^{2} + 14 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{10 \,{\left (a^{4} x^{3} - a^{3} x^{2}\right )}}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}} \sqrt{\frac{a x - 1}{a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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