Optimal. Leaf size=18 \[ \frac {2 \sqrt {-x+x^4}}{3 x^2} \]
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Rubi [A]
time = 0.01, 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 = {2039}
\begin {gather*} \frac {2 \sqrt {x^4-x}}{3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {-x+x^4}} \, dx &=\frac {2 \sqrt {-x+x^4}}{3 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x \left (-1+x^3\right )}}{3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.41, size = 15, normalized size = 0.83
method | result | size |
default | \(\frac {2 \sqrt {x^{4}-x}}{3 x^{2}}\) | \(15\) |
trager | \(\frac {2 \sqrt {x^{4}-x}}{3 x^{2}}\) | \(15\) |
elliptic | \(\frac {2 \sqrt {x^{4}-x}}{3 x^{2}}\) | \(15\) |
risch | \(\frac {\frac {2 x^{3}}{3}-\frac {2}{3}}{x \sqrt {x \left (x^{3}-1\right )}}\) | \(20\) |
gosper | \(\frac {2 \left (-1+x \right ) \left (x^{2}+x +1\right )}{3 x \sqrt {x^{4}-x}}\) | \(24\) |
meijerg | \(-\frac {2 \sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, \sqrt {-x^{3}+1}}{3 \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, x^{\frac {3}{2}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 25, normalized size = 1.39 \begin {gather*} \frac {2 \, {\left (x^{4} - x\right )}}{3 \, \sqrt {x^{2} + x + 1} \sqrt {x - 1} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 14, normalized size = 0.78 \begin {gather*} \frac {2 \, \sqrt {x^{4} - x}}{3 \, x^{2}} \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}{x^{2} \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 11, normalized size = 0.61 \begin {gather*} \frac {2}{3} \, \sqrt {-\frac {1}{x^{3}} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 14, normalized size = 0.78 \begin {gather*} \frac {2\,\sqrt {x^4-x}}{3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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