Optimal. Leaf size=213 \[ \frac{61 x \sqrt [4]{1-\frac{1}{a x}}}{24 a^2 \sqrt [4]{\frac{1}{a x}+1}}+\frac{287 \sqrt [4]{1-\frac{1}{a x}}}{24 a^3 \sqrt [4]{\frac{1}{a x}+1}}+\frac{55 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}-\frac{55 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{x^3 \sqrt [4]{1-\frac{1}{a x}}}{3 \sqrt [4]{\frac{1}{a x}+1}}-\frac{13 x^2 \sqrt [4]{1-\frac{1}{a x}}}{12 a \sqrt [4]{\frac{1}{a x}+1}} \]
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Rubi [A] time = 0.110327, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6171, 98, 151, 155, 12, 93, 298, 203, 206} \[ \frac{61 x \sqrt [4]{1-\frac{1}{a x}}}{24 a^2 \sqrt [4]{\frac{1}{a x}+1}}+\frac{287 \sqrt [4]{1-\frac{1}{a x}}}{24 a^3 \sqrt [4]{\frac{1}{a x}+1}}+\frac{55 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}-\frac{55 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{x^3 \sqrt [4]{1-\frac{1}{a x}}}{3 \sqrt [4]{\frac{1}{a x}+1}}-\frac{13 x^2 \sqrt [4]{1-\frac{1}{a x}}}{12 a \sqrt [4]{\frac{1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 98
Rule 151
Rule 155
Rule 12
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int e^{-\frac{5}{2} \coth ^{-1}(a x)} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/4}}{x^4 \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt [4]{1-\frac{1}{a x}} x^3}{3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{13}{2 a}-\frac{6 x}{a^2}}{x^3 \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{13 \sqrt [4]{1-\frac{1}{a x}} x^2}{12 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^3}{3 \sqrt [4]{1+\frac{1}{a x}}}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{\frac{61}{4 a^2}-\frac{13 x}{a^3}}{x^2 \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{61 \sqrt [4]{1-\frac{1}{a x}} x}{24 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{13 \sqrt [4]{1-\frac{1}{a x}} x^2}{12 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^3}{3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{\frac{165}{8 a^3}-\frac{61 x}{4 a^4}}{x \left (1-\frac{x}{a}\right )^{3/4} \left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{287 \sqrt [4]{1-\frac{1}{a x}}}{24 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{61 \sqrt [4]{1-\frac{1}{a x}} x}{24 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{13 \sqrt [4]{1-\frac{1}{a x}} x^2}{12 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^3}{3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{1}{3} a \operatorname{Subst}\left (\int \frac{165}{16 a^4 x \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{287 \sqrt [4]{1-\frac{1}{a x}}}{24 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{61 \sqrt [4]{1-\frac{1}{a x}} x}{24 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{13 \sqrt [4]{1-\frac{1}{a x}} x^2}{12 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^3}{3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{55 \operatorname{Subst}\left (\int \frac{1}{x \left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{16 a^3}\\ &=\frac{287 \sqrt [4]{1-\frac{1}{a x}}}{24 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{61 \sqrt [4]{1-\frac{1}{a x}} x}{24 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{13 \sqrt [4]{1-\frac{1}{a x}} x^2}{12 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^3}{3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{55 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{4 a^3}\\ &=\frac{287 \sqrt [4]{1-\frac{1}{a x}}}{24 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{61 \sqrt [4]{1-\frac{1}{a x}} x}{24 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{13 \sqrt [4]{1-\frac{1}{a x}} x^2}{12 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^3}{3 \sqrt [4]{1+\frac{1}{a x}}}-\frac{55 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{55 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}\\ &=\frac{287 \sqrt [4]{1-\frac{1}{a x}}}{24 a^3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{61 \sqrt [4]{1-\frac{1}{a x}} x}{24 a^2 \sqrt [4]{1+\frac{1}{a x}}}-\frac{13 \sqrt [4]{1-\frac{1}{a x}} x^2}{12 a \sqrt [4]{1+\frac{1}{a x}}}+\frac{\sqrt [4]{1-\frac{1}{a x}} x^3}{3 \sqrt [4]{1+\frac{1}{a x}}}+\frac{55 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}-\frac{55 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}\\ \end{align*}
Mathematica [C] time = 8.43072, size = 389, normalized size = 1.83 \[ -\frac{e^{-\frac{5}{2} \coth ^{-1}(a x)} \left (256 e^{4 \coth ^{-1}(a x)} \left (626 e^{2 \coth ^{-1}(a x)}+221 e^{4 \coth ^{-1}(a x)}+437\right ) \text{HypergeometricPFQ}\left (\left \{\frac{3}{4},2,2,2\right \},\left \{1,1,\frac{15}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )+2048 e^{4 \coth ^{-1}(a x)} \left (30 e^{2 \coth ^{-1}(a x)}+13 e^{4 \coth ^{-1}(a x)}+17\right ) \text{HypergeometricPFQ}\left (\left \{\frac{3}{4},2,2,2,2\right \},\left \{1,1,1,\frac{15}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )+4096 e^{4 \coth ^{-1}(a x)} \text{HypergeometricPFQ}\left (\left \{\frac{3}{4},2,2,2,2,2\right \},\left \{1,1,1,1,\frac{15}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )+8192 e^{6 \coth ^{-1}(a x)} \text{HypergeometricPFQ}\left (\left \{\frac{3}{4},2,2,2,2,2\right \},\left \{1,1,1,1,\frac{15}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )+4096 e^{8 \coth ^{-1}(a x)} \text{HypergeometricPFQ}\left (\left \{\frac{3}{4},2,2,2,2,2\right \},\left \{1,1,1,1,\frac{15}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )+824824 e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},e^{2 \coth ^{-1}(a x)}\right )+248094 e^{4 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},e^{2 \coth ^{-1}(a x)}\right )-85624 e^{6 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},e^{2 \coth ^{-1}(a x)}\right )-2387 e^{8 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},e^{2 \coth ^{-1}(a x)}\right )+818741 \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},e^{2 \coth ^{-1}(a x)}\right )-1530529 e^{2 \coth ^{-1}(a x)}-266035 e^{4 \coth ^{-1}(a x)}+7161 e^{6 \coth ^{-1}(a x)}-818741\right )}{44352 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54513, size = 279, normalized size = 1.31 \begin{align*} -\frac{1}{48} \, a{\left (\frac{4 \,{\left (137 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} - 174 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} + 69 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{\frac{3 \,{\left (a x - 1\right )} a^{4}}{a x + 1} - \frac{3 \,{\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac{330 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{4}} + \frac{165 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{4}} - \frac{165 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{4}} - \frac{384 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59335, size = 284, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 61 \, a x + 287\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 330 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 165 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + 165 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{48 \, a^{3}} \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 [A] time = 1.18864, size = 259, normalized size = 1.22 \begin{align*} -\frac{1}{48} \, a{\left (\frac{330 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{4}} + \frac{165 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{4}} - \frac{165 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{4}} - \frac{384 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a^{4}} - \frac{4 \,{\left (\frac{174 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{a x + 1} - \frac{137 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{{\left (a x + 1\right )}^{2}} - 69 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}}{a^{4}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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