Optimal. Leaf size=213 \[ -\frac {287 \sqrt [4]{\frac {1}{a x}+1}}{24 a^3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {55 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}+\frac {55 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}+\frac {61 x \sqrt [4]{\frac {1}{a x}+1}}{24 a^2 \sqrt [4]{1-\frac {1}{a x}}}+\frac {x^3 \sqrt [4]{\frac {1}{a x}+1}}{3 \sqrt [4]{1-\frac {1}{a x}}}+\frac {13 x^2 \sqrt [4]{\frac {1}{a x}+1}}{12 a \sqrt [4]{1-\frac {1}{a x}}} \]
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Rubi [A] time = 0.11, 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 = {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 12
Rule 93
Rule 98
Rule 151
Rule 155
Rule 203
Rule 206
Rule 212
Rule 6171
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*}
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Mathematica [A] time = 5.21, size = 137, normalized size = 0.64 \[ \frac {-384 e^{\frac {1}{2} \coth ^{-1}(a x)}+\frac {548 e^{\frac {1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac {400 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}+\frac {128 e^{\frac {1}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}-165 \log \left (1-e^{\frac {1}{2} \coth ^{-1}(a x)}\right )+165 \log \left (e^{\frac {1}{2} \coth ^{-1}(a x)}+1\right )+330 \tan ^{-1}\left (e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{48 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 136, normalized size = 0.64 \[ -\frac {330 \, {\left (a x - 1\right )} \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 165 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 165 \, {\left (a x - 1\right )} \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right ) - 2 \, {\left (8 \, a^{4} x^{4} + 34 \, a^{3} x^{3} + 87 \, a^{2} x^{2} - 226 \, a x - 287\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{48 \, {\left (a^{4} x - a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 192, normalized size = 0.90 \[ -\frac {1}{48} \, a {\left (\frac {330 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{4}} - \frac {165 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{4}} + \frac {165 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{4}} + \frac {384}{a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} - \frac {4 \, {\left (\frac {174 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - \frac {69 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} - 137 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 203, normalized size = 0.95 \[ -\frac {1}{48} \, a {\left (\frac {4 \, {\left (\frac {425 \, {\left (a x - 1\right )}}{a x + 1} - \frac {462 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {165 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 96\right )}}{a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} - 3 \, a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} + 3 \, a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} - a^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} + \frac {330 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{4}} - \frac {165 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{4}} + \frac {165 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 176, normalized size = 0.83 \[ \frac {55\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8\,a^3}-\frac {55\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8\,a^3}-\frac {\frac {77\,{\left (a\,x-1\right )}^2}{2\,{\left (a\,x+1\right )}^2}-\frac {55\,{\left (a\,x-1\right )}^3}{4\,{\left (a\,x+1\right )}^3}-\frac {425\,\left (a\,x-1\right )}{12\,\left (a\,x+1\right )}+8}{a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}-3\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}+3\,a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}-a^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}} \]
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 \[ \int \frac {x^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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