Optimal. Leaf size=18 \[ -\frac {2 \left (x^4+x^2\right )^{5/4}}{5 x^5} \]
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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2014} \begin {gather*} -\frac {2 \left (x^4+x^2\right )^{5/4}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{x^2+x^4}}{x^4} \, dx &=-\frac {2 \left (x^2+x^4\right )^{5/4}}{5 x^5}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2 \left (x^2+1\right ) \sqrt [4]{x^4+x^2}}{5 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2 \left (x^2+x^4\right )^{5/4}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 19, normalized size = 1.06 \begin {gather*} -\frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}{5 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 9, normalized size = 0.50 \begin {gather*} -\frac {2}{5} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 13, normalized size = 0.72
method | result | size |
meijerg | \(-\frac {2 \left (x^{2}+1\right )^{\frac {5}{4}}}{5 x^{\frac {5}{2}}}\) | \(13\) |
gosper | \(-\frac {2 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(20\) |
trager | \(-\frac {2 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(20\) |
risch | \(-\frac {2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{4}+2 x^{2}+1\right )}{5 x^{3} \left (x^{2}+1\right )}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 17, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (x^{3} + x\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}{5 \, x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 29, normalized size = 1.61 \begin {gather*} -\frac {2\,{\left (x^4+x^2\right )}^{1/4}}{5\,x}-\frac {2\,{\left (x^4+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^{2} + 1\right )}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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