Optimal. Leaf size=18 \[ \frac {2 \left (x^6+x^2\right )^{5/4}}{5 x^5} \]
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Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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 = 23, normalized size = 1.28 \begin {gather*} \frac {2 \left (x^4+1\right ) \sqrt [4]{x^6+x^2}}{5 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 18, 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 = 0.47, size = 19, normalized size = 1.06 \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 = 0.09, size = 20, normalized size = 1.11
method | result | size |
gosper | \(\frac {2 \left (x^{4}+1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(20\) |
trager | \(\frac {2 \left (x^{4}+1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(20\) |
meijerg | \(\frac {2 \hypergeom \left (\left [-\frac {5}{8}, -\frac {1}{4}\right ], \left [\frac {3}{8}\right ], -x^{4}\right )}{5 x^{\frac {5}{2}}}+\frac {2 \hypergeom \left (\left [-\frac {1}{4}, \frac {3}{8}\right ], \left [\frac {11}{8}\right ], -x^{4}\right ) x^{\frac {3}{2}}}{3}\) | \(34\) |
risch | \(\frac {2 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \left (x^{8}+2 x^{4}+1\right )}{5 x^{3} \left (x^{4}+1\right )}\) | \(34\) |
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 = 19, normalized size = 1.06 \begin {gather*} \frac {2\,{\left (x^6+x^2\right )}^{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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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