Optimal. Leaf size=261 \[ \frac{19 x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{45 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{8 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a^3 \sqrt{1-\frac{1}{a x}}}+\frac{x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{13 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.316886, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6182, 6180, 98, 151, 156, 63, 208, 206} \[ \frac{19 x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{45 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{8 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a^3 \sqrt{1-\frac{1}{a x}}}+\frac{x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{13 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6180
Rule 98
Rule 151
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^2 \, 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 )^{3/2}}{x^4 \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{-\frac{13}{2 a}-\frac{11 x}{2 a^2}}{x^3 \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{3 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{13 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{\frac{57}{4 a^2}+\frac{39 x}{4 a^3}}{x^2 \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{6 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{19 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{8 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{13 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{-\frac{135}{8 a^3}-\frac{57 x}{8 a^4}}{x \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{6 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{19 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{8 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{13 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^4 \sqrt{1-\frac{1}{a x}}}-\frac{\left (45 \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 )}{16 a^3 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{19 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{8 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{13 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{a^3 \sqrt{1-\frac{1}{a x}}}-\frac{\left (45 \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 )}{8 a^2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{19 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x}{8 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{13 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^2}{12 a \sqrt{1-\frac{1}{a x}}}+\frac{\sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}} x^3}{3 \sqrt{1-\frac{1}{a x}}}+\frac{45 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{8 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{a^3 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.719492, size = 244, normalized size = 0.93 \[ \frac{\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^2 x^2+26 a x+57\right ) \sqrt{c-\frac{c}{a x}}}{a x-1}+135 \sqrt{c} \log \left (2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )-96 \sqrt{2} \sqrt{c} \log \left (2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (3 a^2 x^2-2 a x-1\right )\right )-135 \sqrt{c} \log (1-a x)+96 \sqrt{2} \sqrt{c} \log \left ((a x-1)^2\right )}{48 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.174, size = 202, normalized size = 0.8 \begin{align*}{\frac{ \left ( ax-1 \right ) x}{48\,ax+48}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 16\,{a}^{7/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}{x}^{2}+52\,{a}^{5/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}x+114\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}-96\,\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) \sqrt{a}+135\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{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{\sqrt{c - \frac{c}{a x}} x^{2}}{\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\sqrt{c - \frac{c}{a x}} x^{2}}{\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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