Optimal. Leaf size=20 \[ \frac {2 \left (x^6-x^2\right )^{5/4}}{5 x^5} \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1590} \begin {gather*} \frac {2 \left (x^6-x^2\right )^{5/4}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin {align*} \int \frac {\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}}{x^4} \, dx &=\frac {2 \left (-x^2+x^6\right )^{5/4}}{5 x^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2 \left (x^4-1\right )\right )^{5/4}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \left (-x^2+x^6\right )^{5/4}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 21, normalized size = 1.05 \begin {gather*} \frac {2 \, {\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{5 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 22, normalized size = 1.10
method | result | size |
trager | \(\frac {2 \left (x^{4}-1\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(22\) |
gosper | \(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(28\) |
risch | \(\frac {2 \left (x^{2} \left (x^{4}-1\right )\right )^{\frac {1}{4}} \left (x^{8}-2 x^{4}+1\right )}{5 x^{3} \left (x^{4}-1\right )}\) | \(34\) |
meijerg | \(-\frac {2 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {5}{8}, -\frac {1}{4}\right ], \left [\frac {3}{8}\right ], x^{4}\right )}{5 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} x^{\frac {5}{2}}}+\frac {2 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {3}{8}\right ], \left [\frac {11}{8}\right ], x^{4}\right ) x^{\frac {3}{2}}}{3 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 21, normalized size = 1.05 \begin {gather*} \frac {2\,\left (x^4-1\right )\,{\left (x^6-x^2\right )}^{1/4}}{5\,x^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 \frac {\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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