Optimal. Leaf size=18 \[ \frac {2 \left (-x+x^4\right )^{3/2}}{9 x^6} \]
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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 \left (x^4-x\right )^{3/2}}{9 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rubi steps
\begin {align*} \int \frac {\sqrt {-x+x^4}}{x^6} \, dx &=\frac {2 \left (-x+x^4\right )^{3/2}}{9 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \left (x \left (-1+x^3\right )\right )^{3/2}}{9 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(29\) vs.
\(2(14)=28\).
time = 1.35, size = 30, normalized size = 1.67
method | result | size |
trager | \(\frac {2 \left (x^{3}-1\right ) \sqrt {x^{4}-x}}{9 x^{5}}\) | \(20\) |
gosper | \(\frac {2 \left (-1+x \right ) \left (x^{2}+x +1\right ) \sqrt {x^{4}-x}}{9 x^{5}}\) | \(24\) |
risch | \(\frac {\frac {2}{9} x^{6}-\frac {4}{9} x^{3}+\frac {2}{9}}{x^{4} \sqrt {x \left (x^{3}-1\right )}}\) | \(25\) |
default | \(-\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {2 \sqrt {x^{4}-x}}{9 x^{2}}\) | \(30\) |
elliptic | \(-\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {2 \sqrt {x^{4}-x}}{9 x^{2}}\) | \(30\) |
meijerg | \(-\frac {2 \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, \left (-x^{3}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, x^{\frac {9}{2}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 25, normalized size = 1.39 \begin {gather*} \frac {2 \, {\left (x^{4} - x\right )} \sqrt {x^{2} + x + 1} \sqrt {x - 1}}{9 \, x^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 19, normalized size = 1.06 \begin {gather*} \frac {2 \, \sqrt {x^{4} - x} {\left (x^{3} - 1\right )}}{9 \, x^{5}} \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 {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x^{6}}\, 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}{9} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 19, normalized size = 1.06 \begin {gather*} \frac {2\,\sqrt {x^4-x}\,\left (x^3-1\right )}{9\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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