Optimal. Leaf size=213 \[ \frac{61 x \sqrt [4]{\frac{1}{a x}+1}}{24 a^2 \sqrt [4]{1-\frac{1}{a x}}}-\frac{287 \sqrt [4]{\frac{1}{a x}+1}}{24 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{55 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{55 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{x^3 \sqrt [4]{\frac{1}{a x}+1}}{3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{13 x^2 \sqrt [4]{\frac{1}{a x}+1}}{12 a \sqrt [4]{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.112202, 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, 212, 206, 203} \[ \frac{61 x \sqrt [4]{\frac{1}{a x}+1}}{24 a^2 \sqrt [4]{1-\frac{1}{a x}}}-\frac{287 \sqrt [4]{\frac{1}{a x}+1}}{24 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{55 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{55 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{x^3 \sqrt [4]{\frac{1}{a x}+1}}{3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{13 x^2 \sqrt [4]{\frac{1}{a x}+1}}{12 a \sqrt [4]{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 98
Rule 151
Rule 155
Rule 12
Rule 93
Rule 212
Rule 206
Rule 203
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 )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, 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 )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, 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 )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, 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 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, 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{55 \operatorname{Subst}\left (\int \frac{1}{x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, 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{1}{-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 = 9.0558, size = 441, normalized size = 2.07 \[ -\frac{8 e^{\frac{9}{2} \coth ^{-1}(a x)} \left (\frac{e^{2 \coth ^{-1}(a x)} \left (1906 e^{2 \coth ^{-1}(a x)}+821 e^{4 \coth ^{-1}(a x)}+1117\right ) \text{HypergeometricPFQ}\left (\left \{2,2,2,\frac{13}{4}\right \},\left \{1,1,\frac{25}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )}{3094}+\frac{4 e^{2 \coth ^{-1}(a x)} \left (50 e^{2 \coth ^{-1}(a x)}+23 e^{4 \coth ^{-1}(a x)}+27\right ) \text{HypergeometricPFQ}\left (\left \{2,2,2,2,\frac{13}{4}\right \},\left \{1,1,1,\frac{25}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )}{1547}+\frac{8 e^{2 \coth ^{-1}(a x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{13}{4}\right \},\left \{1,1,1,1,\frac{25}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )}{1547}+\frac{16 e^{4 \coth ^{-1}(a x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{13}{4}\right \},\left \{1,1,1,1,\frac{25}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )}{1547}+\frac{8 e^{6 \coth ^{-1}(a x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{13}{4}\right \},\left \{1,1,1,1,\frac{25}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )}{1547}+\frac{899079}{512} e^{-8 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},e^{2 \coth ^{-1}(a x)}\right )+\frac{60267}{64} e^{-6 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},e^{2 \coth ^{-1}(a x)}\right )-\frac{382227}{256} e^{-4 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},e^{2 \coth ^{-1}(a x)}\right )-\frac{40827}{64} e^{-2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},e^{2 \coth ^{-1}(a x)}\right )+\frac{133407}{512} \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},e^{2 \coth ^{-1}(a x)}\right )-\frac{899079}{512} e^{-8 \coth ^{-1}(a x)}-\frac{3309759 e^{-6 \coth ^{-1}(a x)}}{2560}+\frac{8521937 e^{-4 \coth ^{-1}(a x)}}{7680}+\frac{69571361 e^{-2 \coth ^{-1}(a x)}}{99840}-\frac{653}{390} e^{2 \coth ^{-1}(a x)}-\frac{27653}{195}\right )}{9 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.322, 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 = 1.55226, size = 274, normalized size = 1.29 \begin{align*} -\frac{1}{48} \, a{\left (\frac{4 \,{\left (\frac{425 \,{\left (a x - 1\right )}}{a x + 1} - \frac{462 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{165 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 96\right )}}{a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{4}} - 3 \, a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} + 3 \, a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} - a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{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}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58515, size = 358, normalized size = 1.68 \begin{align*} -\frac{330 \,{\left (a x - 1\right )} \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 165 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + 165 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right ) - 2 \,{\left (8 \, a^{4} x^{4} + 34 \, a^{3} x^{3} + 87 \, a^{2} x^{2} - 226 \, a x - 287\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{48 \,{\left (a^{4} x - 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}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18243, 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}{a^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} - \frac{4 \,{\left (\frac{174 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{a x + 1} - \frac{69 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{2}} - 137 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{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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