Optimal. Leaf size=147 \[ \frac{19 x \sqrt{c-\frac{c}{a x}}}{8 a^2}-\frac{45 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{8 a^3}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a^3}+\frac{1}{3} x^3 \sqrt{c-\frac{c}{a x}}-\frac{13 x^2 \sqrt{c-\frac{c}{a x}}}{12 a} \]
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Rubi [A] time = 0.451411, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {6167, 6133, 25, 514, 446, 98, 151, 156, 63, 208} \[ \frac{19 x \sqrt{c-\frac{c}{a x}}}{8 a^2}-\frac{45 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{8 a^3}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a^3}+\frac{1}{3} x^3 \sqrt{c-\frac{c}{a x}}-\frac{13 x^2 \sqrt{c-\frac{c}{a x}}}{12 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 446
Rule 98
Rule 151
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx\\ &=-\int \frac{\sqrt{c-\frac{c}{a x}} x^2 (1-a x)}{1+a x} \, dx\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{3/2} x^3}{1+a x} \, dx}{c}\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{3/2} x^2}{a+\frac{1}{x}} \, dx}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{x^4 (a+x)} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3+\frac{\operatorname{Subst}\left (\int \frac{\frac{13 c^2}{2}-\frac{11 c^2 x}{2 a}}{x^3 (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{3 c}\\ &=-\frac{13 \sqrt{c-\frac{c}{a x}} x^2}{12 a}+\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3-\frac{\operatorname{Subst}\left (\int \frac{\frac{57 c^3}{4}-\frac{39 c^3 x}{4 a}}{x^2 (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{6 a c^2}\\ &=\frac{19 \sqrt{c-\frac{c}{a x}} x}{8 a^2}-\frac{13 \sqrt{c-\frac{c}{a x}} x^2}{12 a}+\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3+\frac{\operatorname{Subst}\left (\int \frac{\frac{135 c^4}{8}-\frac{57 c^4 x}{8 a}}{x (a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{6 a^2 c^3}\\ &=\frac{19 \sqrt{c-\frac{c}{a x}} x}{8 a^2}-\frac{13 \sqrt{c-\frac{c}{a x}} x^2}{12 a}+\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3+\frac{(45 c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a^3}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^3}\\ &=\frac{19 \sqrt{c-\frac{c}{a x}} x}{8 a^2}-\frac{13 \sqrt{c-\frac{c}{a x}} x^2}{12 a}+\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3-\frac{45 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{8 a^2}+\frac{8 \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{a^2}\\ &=\frac{19 \sqrt{c-\frac{c}{a x}} x}{8 a^2}-\frac{13 \sqrt{c-\frac{c}{a x}} x^2}{12 a}+\frac{1}{3} \sqrt{c-\frac{c}{a x}} x^3-\frac{45 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{8 a^3}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.107813, size = 108, normalized size = 0.73 \[ \frac{a x \left (8 a^2 x^2-26 a x+57\right ) \sqrt{c-\frac{c}{a x}}-135 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )+96 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )}{24 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.158, size = 237, normalized size = 1.6 \begin{align*}{\frac{x}{48}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 16\, \left ( a{x}^{2}-x \right ) ^{3/2}{a}^{7/2}\sqrt{{a}^{-1}}-36\,\sqrt{a{x}^{2}-x}{a}^{7/2}\sqrt{{a}^{-1}}x+18\,\sqrt{a{x}^{2}-x}{a}^{5/2}\sqrt{{a}^{-1}}+96\,\sqrt{ \left ( ax-1 \right ) x}{a}^{5/2}\sqrt{{a}^{-1}}-96\,{a}^{3/2}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1}{ax+1}} \right ) -144\,{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{a}^{2} \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{a}^{-{\frac{9}{2}}}{\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 )} \sqrt{c - \frac{c}{a x}} x^{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 = 1.64023, size = 612, normalized size = 4.16 \begin{align*} \left [\frac{96 \, \sqrt{2} \sqrt{c} \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 ) + 2 \,{\left (8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt{\frac{a c x - c}{a x}} + 135 \, \sqrt{c} \log \left (-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right )}{48 \, a^{3}}, -\frac{96 \, \sqrt{2} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right ) -{\left (8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt{\frac{a c x - c}{a x}} - 135 \, \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right )}{24 \, a^{3}}\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{x^{2} \sqrt{- c \left (-1 + \frac{1}{a x}\right )} \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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