Optimal. Leaf size=16 \[ \frac {\left (x^6-1\right )^{4/3}}{4 x^4} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {449} \begin {gather*} \frac {\left (x^6-1\right )^{4/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx &=\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (x^6-1\right )^{4/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^6\right )^{4/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 13, normalized size = 0.81
method | result | size |
trager | \(\frac {\left (x^{6}-1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
risch | \(\frac {x^{12}-2 x^{6}+1}{4 \left (x^{6}-1\right )^{\frac {2}{3}} x^{4}}\) | \(23\) |
gosper | \(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{4 x^{4}}\) | \(33\) |
meijerg | \(\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right ) x^{2}}{2 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}}}-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {1}{3}\right ], \left [\frac {1}{3}\right ], x^{6}\right )}{4 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {1}{3}} x^{4}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 38, normalized size = 2.38 \begin {gather*} \frac {{\left (x^{6} - 1\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^6-1\right )}^{4/3}}{4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.01, size = 71, normalized size = 4.44 \begin {gather*} \frac {x^{2} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} - \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{4} \Gamma \left (\frac {1}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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