Optimal. Leaf size=14 \[ \frac {4 \sqrt [4]{x^5-1}}{x} \]
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Rubi [A] time = 0.01, antiderivative size = 14, 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 {4 \sqrt [4]{x^5-1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin {align*} \int \frac {4+x^5}{x^2 \left (-1+x^5\right )^{3/4}} \, dx &=\frac {4 \sqrt [4]{-1+x^5}}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{x^5-1}}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 14, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{-1+x^5}}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 12, normalized size = 0.86 \begin {gather*} \frac {4 \, {\left (x^{5} - 1\right )}^{\frac {1}{4}}}{x} \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 {x^{5} + 4}{{\left (x^{5} - 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 13, normalized size = 0.93
method | result | size |
trager | \(\frac {4 \left (x^{5}-1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
risch | \(\frac {4 \left (x^{5}-1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
gosper | \(\frac {4 \left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}{x \left (x^{5}-1\right )^{\frac {3}{4}}}\) | \(28\) |
meijerg | \(\frac {\left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {3}{4}, \frac {4}{5}\right ], \left [\frac {9}{5}\right ], x^{5}\right ) x^{4}}{4 \mathrm {signum}\left (x^{5}-1\right )^{\frac {3}{4}}}-\frac {4 \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [-\frac {1}{5}, \frac {3}{4}\right ], \left [\frac {4}{5}\right ], x^{5}\right )}{\mathrm {signum}\left (x^{5}-1\right )^{\frac {3}{4}} x}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 24, normalized size = 1.71 \begin {gather*} \frac {4 \, {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 12, normalized size = 0.86 \begin {gather*} \frac {4\,{\left (x^5-1\right )}^{1/4}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.69, size = 68, normalized size = 4.86 \begin {gather*} \frac {x^{4} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {4}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {4}{5} \\ \frac {9}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 \Gamma \left (\frac {9}{5}\right )} - \frac {4 e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{5}, \frac {3}{4} \\ \frac {4}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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