Optimal. Leaf size=196 \[ \frac{49 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{15 a \sqrt{c-\frac{c}{a x}}}+\frac{31 c^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}}{15 a}+\frac{7 c^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{3/2}}{5 a}-\frac{5 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{a}+c x \sqrt{1-\frac{1}{a^2 x^2}} \left (c-\frac{c}{a x}\right )^{5/2} \]
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Rubi [A] time = 0.148126, antiderivative size = 221, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6182, 6179, 97, 153, 147, 63, 208} \[ \frac{x \left (a-\frac{1}{x}\right )^3 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{7 \left (a-\frac{1}{x}\right )^2 \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{\left (80 a-\frac{31}{x}\right ) \sqrt{\frac{1}{a x}+1} \left (c-\frac{c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}-\frac{5 \left (c-\frac{c}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \left (1-\frac{1}{a x}\right )^{7/2}} \]
Warning: Unable to verify antiderivative.
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Rule 6182
Rule 6179
Rule 97
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 )^3 \sqrt{1+\frac{x}{a}}}{x^2} \, 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{5}{2 a}-\frac{7 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{7 \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{25}{4 a^2}-\frac{31 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{31}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{7 \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 (5 \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{31}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{7 \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 (5 \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{31}{x}\right ) \sqrt{1+\frac{1}{a x}} \left (c-\frac{c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac{1}{a x}\right )^{7/2}}+\frac{7 \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{5 \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 = 0.0818589, size = 101, normalized size = 0.52 \[ \frac{c^3 \sqrt{c-\frac{c}{a x}} \left (\sqrt{\frac{1}{a x}+1} \left (15 a^3 x^3+56 a^2 x^2-28 a x+6\right )-75 a^2 x^2 \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )\right )}{15 a^3 x^2 \sqrt{1-\frac{1}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.18, size = 149, normalized size = 0.8 \begin{align*} -{\frac{{c}^{3}}{30\,{x}^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -30\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}+75\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){x}^{3}{a}^{3}-112\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}+56\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-12\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{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 \frac{{\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 = 2.2213, size = 861, normalized size = 4.39 \begin{align*} \left [\frac{75 \,{\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 (15 \, a^{4} c^{3} x^{4} + 71 \, a^{3} c^{3} x^{3} + 28 \, a^{2} c^{3} x^{2} - 22 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{60 \,{\left (a^{4} x^{3} - a^{3} x^{2}\right )}}, \frac{75 \,{\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 (15 \, a^{4} c^{3} x^{4} + 71 \, a^{3} c^{3} x^{3} + 28 \, a^{2} c^{3} x^{2} - 22 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{30 \,{\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 \frac{{\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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