Optimal. Leaf size=13 \[ \frac {2}{5} \left (-1+x^2\right )^{5/4} \]
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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267}
\begin {gather*} \frac {2}{5} \left (x^2-1\right )^{5/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rubi steps
\begin {align*} \int x \sqrt [4]{-1+x^2} \, dx &=\frac {2}{5} \left (-1+x^2\right )^{5/4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {2}{5} \left (-1+x^2\right )^{5/4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 10, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {2 \left (x^{2}-1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
default | \(\frac {2 \left (x^{2}-1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
risch | \(\frac {2 \left (x^{2}-1\right )^{\frac {5}{4}}}{5}\) | \(10\) |
gosper | \(\frac {2 \left (1+x \right ) \left (-1+x \right ) \left (x^{2}-1\right )^{\frac {1}{4}}}{5}\) | \(16\) |
trager | \(\left (\frac {2 x^{2}}{5}-\frac {2}{5}\right ) \left (x^{2}-1\right )^{\frac {1}{4}}\) | \(16\) |
meijerg | \(\frac {\mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{2} \hypergeom \left (\left [-\frac {1}{4}, 1\right ], \left [2\right ], x^{2}\right )}{2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{5} \, {\left (x^{2} - 1\right )}^{\frac {5}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{5} \, {\left (x^{2} - 1\right )}^{\frac {5}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (10) = 20\).
time = 0.07, size = 26, normalized size = 2.00 \begin {gather*} \frac {2 x^{2} \sqrt [4]{x^{2} - 1}}{5} - \frac {2 \sqrt [4]{x^{2} - 1}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 9, normalized size = 0.69 \begin {gather*} \frac {2}{5} \, {\left (x^{2} - 1\right )}^{\frac {5}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 9, normalized size = 0.69 \begin {gather*} \frac {2\,{\left (x^2-1\right )}^{5/4}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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