Optimal. Leaf size=164 \[ \frac{21 c^3 \sqrt{c-\frac{c}{a x}}}{a}+\frac{5 c^2 \left (c-\frac{c}{a x}\right )^{3/2}}{3 a}+\frac{11 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}-\frac{32 \sqrt{2} c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a}-\frac{3 c \left (c-\frac{c}{a x}\right )^{5/2}}{5 a}-x \left (c-\frac{c}{a x}\right )^{7/2} \]
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Rubi [A] time = 0.226423, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {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 6133
Rule 25
Rule 514
Rule 375
Rule 98
Rule 154
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \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 = 0.17077, size = 125, normalized size = 0.76 \[ \frac{c^3 \left (-15 a^3 x^3+376 a^2 x^2-52 a x+6\right ) \sqrt{c-\frac{c}{a x}}}{15 a^3 x^2}+\frac{11 c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a}-\frac{32 \sqrt{2} c^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.128, size = 281, normalized size = 1.7 \begin{align*} -{\frac{{c}^{3}}{30\,{x}^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -1110\,\sqrt{a{x}^{2}-x}{a}^{7/2}\sqrt{{a}^{-1}}{x}^{4}+480\,{a}^{7/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}{x}^{4}+660\,{a}^{5/2} \left ( a{x}^{2}-x \right ) ^{3/2}{x}^{2}\sqrt{{a}^{-1}}+555\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{x}^{4}{a}^{3}-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}-720\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{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^{2} x^{2} - 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}{{\left (a x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97427, size = 741, normalized size = 4.52 \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{4 c^{3} \sqrt{c - \frac{c}{a x}}}{a x + 1}\, dx - \int \frac{6 c^{3} \sqrt{c - \frac{c}{a x}}}{a^{2} x^{2} + a x}\, dx - \int - \frac{4 c^{3} \sqrt{c - \frac{c}{a x}}}{a^{3} x^{3} + a^{2} x^{2}}\, dx - \int \frac{c^{3} \sqrt{c - \frac{c}{a x}}}{a^{4} x^{4} + a^{3} x^{3}}\, dx - \int \frac{a c^{3} x \sqrt{c - \frac{c}{a x}}}{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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