Optimal. Leaf size=213 \[ \frac {287 \sqrt [4]{1-\frac {1}{a x}}}{24 a^3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {61 \sqrt [4]{1-\frac {1}{a x}} x}{24 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {13 \sqrt [4]{1-\frac {1}{a x}} x^2}{12 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^3}{3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {55 \text {ArcTan}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}-\frac {55 \tanh ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3} \]
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Rubi [A]
time = 0.07, antiderivative size = 213, normalized size of antiderivative = 1.00, 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 = {6306, 100,
156, 160, 12, 95, 304, 209, 212} \begin {gather*} \frac {55 \text {ArcTan}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\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]{\frac {1}{a x}+1}}-\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 {61 x \sqrt [4]{1-\frac {1}{a x}}}{24 a^2 \sqrt [4]{\frac {1}{a x}+1}}+\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 160
Rule 209
Rule 212
Rule 304
Rule 6306
Rubi steps
\begin {align*} \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^2 \, dx &=-\text {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} \text {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} \text {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} \text {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 \text {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 \text {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 \text {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 \text {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 \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 5.66, size = 389, normalized size = 1.83 \begin {gather*} -\frac {e^{-\frac {5}{2} \coth ^{-1}(a x)} \left (-818741-1530529 e^{2 \coth ^{-1}(a x)}-266035 e^{4 \coth ^{-1}(a x)}+7161 e^{6 \coth ^{-1}(a x)}+818741 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 \coth ^{-1}(a x)}\right )+824824 e^{2 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 \coth ^{-1}(a x)}\right )+248094 e^{4 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 \coth ^{-1}(a x)}\right )-85624 e^{6 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 \coth ^{-1}(a x)}\right )-2387 e^{8 \coth ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 \coth ^{-1}(a x)}\right )+256 e^{4 \coth ^{-1}(a x)} \left (437+626 e^{2 \coth ^{-1}(a x)}+221 e^{4 \coth ^{-1}(a x)}\right ) \, _4F_3\left (\frac {3}{4},2,2,2;1,1,\frac {15}{4};e^{2 \coth ^{-1}(a x)}\right )+2048 e^{4 \coth ^{-1}(a x)} \left (17+30 e^{2 \coth ^{-1}(a x)}+13 e^{4 \coth ^{-1}(a x)}\right ) \, _5F_4\left (\frac {3}{4},2,2,2,2;1,1,1,\frac {15}{4};e^{2 \coth ^{-1}(a x)}\right )+4096 e^{4 \coth ^{-1}(a x)} \, _6F_5\left (\frac {3}{4},2,2,2,2,2;1,1,1,1,\frac {15}{4};e^{2 \coth ^{-1}(a x)}\right )+8192 e^{6 \coth ^{-1}(a x)} \, _6F_5\left (\frac {3}{4},2,2,2,2,2;1,1,1,1,\frac {15}{4};e^{2 \coth ^{-1}(a x)}\right )+4096 e^{8 \coth ^{-1}(a x)} \, _6F_5\left (\frac {3}{4},2,2,2,2,2;1,1,1,1,\frac {15}{4};e^{2 \coth ^{-1}(a x)}\right )\right )}{44352 a^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x^{2} \left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 207, normalized size = 0.97 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 103, normalized size = 0.48 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 192, normalized size = 0.90 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 181, normalized size = 0.85 \begin {gather*} \frac {\frac {23\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{4}-\frac {29\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{2}+\frac {137\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}}{12}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\left (a\,x-1\right )}{a\,x+1}}+\frac {8\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{a^3}-\frac {55\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8\,a^3}+\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )\,55{}\mathrm {i}}{8\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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