Optimal. Leaf size=303 \[ \frac{223 x^2 \sqrt{c-\frac{c}{a x}}}{96 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{1115 x \sqrt{c-\frac{c}{a x}}}{192 a^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{1115 \sqrt{c-\frac{c}{a x}}}{64 a^4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{1115 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{64 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{x^4 \sqrt{c-\frac{c}{a x}}}{4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{25 x^3 \sqrt{c-\frac{c}{a x}}}{24 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.318334, antiderivative size = 306, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6182, 6180, 89, 78, 51, 63, 208} \[ \frac{1115 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{96 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{223 x^2 \sqrt{c-\frac{c}{a x}}}{24 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{1115 x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{64 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{1115 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{64 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{x^4 \sqrt{c-\frac{c}{a x}}}{4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{25 x^3 \sqrt{c-\frac{c}{a x}}}{24 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6180
Rule 89
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^3 \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^3 \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^5 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x^4}{4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{-\frac{25}{2 a}+\frac{4 x}{a^2}}{x^4 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{4 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{25 \sqrt{c-\frac{c}{a x}} x^3}{24 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^4}{4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (223 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{48 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{223 \sqrt{c-\frac{c}{a x}} x^2}{24 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{25 \sqrt{c-\frac{c}{a x}} x^3}{24 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^4}{4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (1115 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{48 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{223 \sqrt{c-\frac{c}{a x}} x^2}{24 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{1115 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{96 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{25 \sqrt{c-\frac{c}{a x}} x^3}{24 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^4}{4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\left (1115 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{64 a^3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{1115 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{64 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{223 \sqrt{c-\frac{c}{a x}} x^2}{24 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{1115 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{96 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{25 \sqrt{c-\frac{c}{a x}} x^3}{24 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^4}{4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (1115 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{128 a^4 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{1115 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{64 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{223 \sqrt{c-\frac{c}{a x}} x^2}{24 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{1115 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{96 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{25 \sqrt{c-\frac{c}{a x}} x^3}{24 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^4}{4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{\left (1115 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{64 a^3 \sqrt{1-\frac{1}{a x}}}\\ &=-\frac{1115 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{64 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{223 \sqrt{c-\frac{c}{a x}} x^2}{24 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{1115 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{96 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{25 \sqrt{c-\frac{c}{a x}} x^3}{24 a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} x^4}{4 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{1115 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{64 a^4 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.585685, size = 167, normalized size = 0.55 \[ \frac{\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (48 a^4 x^4-200 a^3 x^3+446 a^2 x^2-1115 a x-3345\right ) \sqrt{c-\frac{c}{a x}}}{a^2 x^2-1}+3345 \sqrt{c} \log \left (2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )-3345 \sqrt{c} \log (1-a x)}{384 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.188, size = 197, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax+1 \right ) x}{384\, \left ( ax-1 \right ) ^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 96\,{a}^{9/2}\sqrt{ \left ( ax+1 \right ) x}{x}^{4}-400\,{a}^{7/2}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}+892\,{a}^{5/2}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}-2230\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}+3345\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) xa-6690\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+3345\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) \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 \sqrt{c - \frac{c}{a x}} x^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9376, size = 792, normalized size = 2.61 \begin{align*} \left [\frac{3345 \,{\left (a x - 1\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 (48 \, a^{5} x^{5} - 200 \, a^{4} x^{4} + 446 \, a^{3} x^{3} - 1115 \, a^{2} x^{2} - 3345 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{768 \,{\left (a^{5} x - a^{4}\right )}}, -\frac{3345 \,{\left (a x - 1\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 (48 \, a^{5} x^{5} - 200 \, a^{4} x^{4} + 446 \, a^{3} x^{3} - 1115 \, a^{2} x^{2} - 3345 \, a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{384 \,{\left (a^{5} x - a^{4}\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 \sqrt{c - \frac{c}{a x}} x^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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