Optimal. Leaf size=23 \[ \frac {2 \left (x^4+x^2\right )^{3/4}}{x \left (x^2+1\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1146, 264} \begin {gather*} \frac {2 x}{\sqrt [4]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 1146
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^2\right ) \sqrt [4]{x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1+x^2\right )^{5/4}} \, dx}{\sqrt [4]{x^2+x^4}}\\ &=\frac {2 x}{\sqrt [4]{x^2+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 0.61 \begin {gather*} \frac {2 x}{\sqrt [4]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2+x^4\right )^{3/4}}{x \left (1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 18, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 9, normalized size = 0.39 \begin {gather*} \frac {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 = 0.06, size = 13, normalized size = 0.57
method | result | size |
gosper | \(\frac {2 x}{\left (x^{4}+x^{2}\right )^{\frac {1}{4}}}\) | \(13\) |
meijerg | \(\frac {2 \sqrt {x}}{\left (x^{2}+1\right )^{\frac {1}{4}}}\) | \(13\) |
risch | \(\frac {2 x}{\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\) | \(15\) |
trager | \(\frac {2 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}+1\right )}\) | \(22\) |
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*} -\frac {2 \, {\left (x^{3} + x\right )}}{3 \, {\left (x^{\frac {5}{2}} + \sqrt {x}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} + \int \frac {4 \, {\left (x^{2} + 1\right )}^{\frac {3}{4}}}{3 \, {\left (x^{\frac {9}{2}} + 2 \, x^{\frac {5}{2}} + \sqrt {x}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 21, normalized size = 0.91 \begin {gather*} \frac {2\,{\left (x^4+x^2\right )}^{3/4}}{x\,\left (x^2+1\right )} \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 {1}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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