Optimal. Leaf size=179 \[ \frac{11 x \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{24 a^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{1}{3} x^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}+\frac{5 x^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{12 a} \]
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Rubi [A] time = 0.0946851, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6171, 99, 151, 12, 93, 212, 206, 203} \[ \frac{11 x \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{24 a^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{1}{3} x^3 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}+\frac{5 x^2 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{\frac{1}{a x}+1}}{12 a} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 99
Rule 151
Rule 12
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int e^{\frac{1}{2} \coth ^{-1}(a x)} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [4]{1+\frac{x}{a}}}{x^4 \sqrt [4]{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{5}{2 a}+\frac{2 x}{a^2}}{x^3 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{12 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3+\frac{1}{6} \operatorname{Subst}\left (\int \frac{-\frac{11}{4 a^2}-\frac{5 x}{2 a^3}}{x^2 \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{24 a^2}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{12 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3-\frac{1}{6} \operatorname{Subst}\left (\int \frac{9}{8 a^3 x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{24 a^2}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{12 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3-\frac{3 \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{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{24 a^2}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{12 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3-\frac{3 \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{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{24 a^2}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{12 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3+\frac{3 \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{3 \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{11 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x}{24 a^2}+\frac{5 \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^2}{12 a}+\frac{1}{3} \left (1-\frac{1}{a x}\right )^{3/4} \sqrt [4]{1+\frac{1}{a x}} x^3+\frac{3 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{8 a^3}+\frac{3 \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 = 5.13347, size = 399, normalized size = 2.23 \[ -\frac{e^{-\frac{7}{2} \coth ^{-1}(a x)} \left (1280 e^{8 \coth ^{-1}(a x)} \left (1346 e^{2 \coth ^{-1}(a x)}+557 e^{4 \coth ^{-1}(a x)}+821\right ) \text{HypergeometricPFQ}\left (\left \{2,2,2,\frac{9}{4}\right \},\left \{1,1,\frac{21}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )+10240 e^{8 \coth ^{-1}(a x)} \left (42 e^{2 \coth ^{-1}(a x)}+19 e^{4 \coth ^{-1}(a x)}+23\right ) \text{HypergeometricPFQ}\left (\left \{2,2,2,2,\frac{9}{4}\right \},\left \{1,1,1,\frac{21}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )+20480 e^{8 \coth ^{-1}(a x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{9}{4}\right \},\left \{1,1,1,1,\frac{21}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )+40960 e^{10 \coth ^{-1}(a x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{9}{4}\right \},\left \{1,1,1,1,\frac{21}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )+20480 e^{12 \coth ^{-1}(a x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{9}{4}\right \},\left \{1,1,1,1,\frac{21}{4}\right \},e^{2 \coth ^{-1}(a x)}\right )+732349800 e^{2 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},e^{2 \coth ^{-1}(a x)}\right )-635067810 e^{4 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},e^{2 \coth ^{-1}(a x)}\right )-384831720 e^{6 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},e^{2 \coth ^{-1}(a x)}\right )+60913125 e^{8 \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},e^{2 \coth ^{-1}(a x)}\right )+1070609085 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},e^{2 \coth ^{-1}(a x)}\right )-946471617 e^{2 \coth ^{-1}(a x)}+369641285 e^{4 \coth ^{-1}(a x)}+351173641 e^{6 \coth ^{-1}(a x)}-23818496 e^{8 \coth ^{-1}(a x)}-1070609085\right )}{1909440 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.135, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\frac{1}{\sqrt [4]{{\frac{ax-1}{ax+1}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48375, size = 252, normalized size = 1.41 \begin{align*} -\frac{1}{48} \, a{\left (\frac{4 \,{\left (9 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{4}} - 6 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{4}} + 29 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{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{18 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{4}} - \frac{9 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{4}} + \frac{9 \, \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 = 1.76249, size = 275, normalized size = 1.54 \begin{align*} \frac{2 \,{\left (8 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 21 \, a x + 11\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}} - 18 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + 9 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) - 9 \, \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt [4]{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21693, size = 232, normalized size = 1.3 \begin{align*} -\frac{1}{48} \, a{\left (\frac{18 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{4}} - \frac{9 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{4}} + \frac{9 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{4}} - \frac{4 \,{\left (\frac{6 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{a x + 1} - \frac{9 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{2}} - 29 \, \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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