Optimal. Leaf size=16 \[ -\frac {2 \sqrt [4]{x^4+x^2}}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1311, 2000, 1146, 271, 264} \begin {gather*} -\frac {2 \sqrt [4]{x^4+x^2}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rule 1146
Rule 1311
Rule 2000
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{x^2+x^4}}{x^2 \left (1+x^2\right )} \, dx &=\int \frac {1}{\left (x^2+x^4\right )^{3/4}} \, dx\\ &=-\frac {2 \sqrt [4]{x^2+x^4}}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt [4]{x^4+x^2}}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt [4]{x^2+x^4}}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 14, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 9, normalized size = 0.56 \begin {gather*} -2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 13, normalized size = 0.81
method | result | size |
meijerg | \(-\frac {2 \left (x^{2}+1\right )^{\frac {1}{4}}}{\sqrt {x}}\) | \(13\) |
gosper | \(-\frac {2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{x}\) | \(15\) |
trager | \(-\frac {2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{x}\) | \(15\) |
risch | \(-\frac {2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 17, normalized size = 1.06 \begin {gather*} -\frac {2 \, {\left (x^{3} + x\right )}}{{\left (x^{2} + 1\right )}^{\frac {3}{4}} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 14, normalized size = 0.88 \begin {gather*} -\frac {2\,{\left (x^4+x^2\right )}^{1/4}}{x} \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^{2} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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