Optimal. Leaf size=163 \[ -\frac{21 c^3 \sqrt{c-\frac{c}{a x}}}{a}-\frac{5 c^2 \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}-\frac{11 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{32 \sqrt{2} c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a}+\frac{3 c \left (c-\frac{c}{a x}\right )^{5/2}}{5 a}+x \left (c-\frac{c}{a x}\right )^{7/2} \]
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Rubi [A] time = 0.280299, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6167, 6133, 25, 514, 375, 98, 154, 156, 63, 208} \[ -\frac{21 c^3 \sqrt{c-\frac{c}{a x}}}{a}-\frac{5 c^2 \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}-\frac{11 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{32 \sqrt{2} c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a}+\frac{3 c \left (c-\frac{c}{a x}\right )^{5/2}}{5 a}+x \left (c-\frac{c}{a x}\right )^{7/2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 98
Rule 154
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{7/2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{7/2} \, dx\\ &=-\int \frac{\left (c-\frac{c}{a x}\right )^{7/2} (1-a x)}{1+a x} \, dx\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{9/2} x}{1+a x} \, dx}{c}\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{9/2}}{a+\frac{1}{x}} \, dx}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{9/2}}{x^2 (a+x)} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\left (c-\frac{c}{a x}\right )^{7/2} x+\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{5/2} \left (\frac{11 c^2}{2}+\frac{3 c^2 x}{2 a}\right )}{x (a+x)} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{3 c \left (c-\frac{c}{a x}\right )^{5/2}}{5 a}+\left (c-\frac{c}{a x}\right )^{7/2} x+\frac{2 \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2} \left (\frac{55 c^3}{4}-\frac{25 c^3 x}{4 a}\right )}{x (a+x)} \, dx,x,\frac{1}{x}\right )}{5 c}\\ &=-\frac{5 c^2 \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+\frac{3 c \left (c-\frac{c}{a x}\right )^{5/2}}{5 a}+\left (c-\frac{c}{a x}\right )^{7/2} x+\frac{4 \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}} \left (\frac{165 c^4}{8}-\frac{315 c^4 x}{8 a}\right )}{x (a+x)} \, dx,x,\frac{1}{x}\right )}{15 c}\\ &=-\frac{21 c^3 \sqrt{c-\frac{c}{a x}}}{a}-\frac{5 c^2 \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+\frac{3 c \left (c-\frac{c}{a x}\right )^{5/2}}{5 a}+\left (c-\frac{c}{a x}\right )^{7/2} x+\frac{8 \operatorname{Subst}\left (\int \frac{\frac{165 c^5}{16}-\frac{795 c^5 x}{16 a}}{x (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{15 c}\\ &=-\frac{21 c^3 \sqrt{c-\frac{c}{a x}}}{a}-\frac{5 c^2 \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+\frac{3 c \left (c-\frac{c}{a x}\right )^{5/2}}{5 a}+\left (c-\frac{c}{a x}\right )^{7/2} x+\frac{\left (11 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}-\frac{\left (32 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{21 c^3 \sqrt{c-\frac{c}{a x}}}{a}-\frac{5 c^2 \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+\frac{3 c \left (c-\frac{c}{a x}\right )^{5/2}}{5 a}+\left (c-\frac{c}{a x}\right )^{7/2} x-\left (11 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )+\left (64 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )\\ &=-\frac{21 c^3 \sqrt{c-\frac{c}{a x}}}{a}-\frac{5 c^2 \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+\frac{3 c \left (c-\frac{c}{a x}\right )^{5/2}}{5 a}+\left (c-\frac{c}{a x}\right )^{7/2} x-\frac{11 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{32 \sqrt{2} c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.260302, size = 125, normalized size = 0.77 \[ \frac{c^3 \left (15 a^3 x^3-376 a^2 x^2+52 a x-6\right ) \sqrt{c-\frac{c}{a x}}}{15 a^3 x^2}-\frac{11 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}+\frac{32 \sqrt{2} c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.168, size = 281, normalized size = 1.7 \begin{align*} -{\frac{{c}^{3}}{30\,{x}^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 1110\,{a}^{7/2}\sqrt{{a}^{-1}}\sqrt{a{x}^{2}-x}{x}^{4}-480\,{a}^{7/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}{x}^{4}+480\,{a}^{5/2}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1}{ax+1}} \right ){x}^{4}-660\,{a}^{5/2}\sqrt{{a}^{-1}} \left ( a{x}^{2}-x \right ) ^{3/2}{x}^{2}-555\,\sqrt{{a}^{-1}}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{4}{a}^{3}+720\,\sqrt{{a}^{-1}}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{4}{a}^{3}+92\,{a}^{3/2} \left ( a{x}^{2}-x \right ) ^{3/2}x\sqrt{{a}^{-1}}-12\, \left ( a{x}^{2}-x \right ) ^{3/2}\sqrt{a}\sqrt{{a}^{-1}} \right ){a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \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 (a x - 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2818, size = 744, normalized size = 4.56 \begin{align*} \left [\frac{480 \, \sqrt{2} a^{2} c^{\frac{7}{2}} x^{2} \log \left (-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 165 \, a^{2} c^{\frac{7}{2}} x^{2} \log \left (-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right ) + 2 \,{\left (15 \, a^{3} c^{3} x^{3} - 376 \, a^{2} c^{3} x^{2} + 52 \, a c^{3} x - 6 \, c^{3}\right )} \sqrt{\frac{a c x - c}{a x}}}{30 \, a^{3} x^{2}}, -\frac{480 \, \sqrt{2} a^{2} \sqrt{-c} c^{3} x^{2} \arctan \left (\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right ) - 165 \, a^{2} \sqrt{-c} c^{3} x^{2} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right ) -{\left (15 \, a^{3} c^{3} x^{3} - 376 \, a^{2} c^{3} x^{2} + 52 \, a c^{3} x - 6 \, c^{3}\right )} \sqrt{\frac{a c x - c}{a x}}}{15 \, a^{3} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- c \left (-1 + \frac{1}{a x}\right )\right )^{\frac{7}{2}} \left (a x - 1\right )}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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