Optimal. Leaf size=20 \[ \frac {4 \left (x^5-x^3\right )^{9/4}}{9 x^9} \]
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Rubi [B] time = 0.20, antiderivative size = 59, normalized size of antiderivative = 2.95, number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2052, 2004, 2032, 365, 364, 2020, 2025} \begin {gather*} \frac {4}{9} \sqrt [4]{x^5-x^3} x-\frac {8 \sqrt [4]{x^5-x^3}}{9 x}+\frac {4 \sqrt [4]{x^5-x^3}}{9 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 2004
Rule 2020
Rule 2025
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^3+x^5}}{x^4} \, dx &=\int \left (\sqrt [4]{-x^3+x^5}-\frac {\sqrt [4]{-x^3+x^5}}{x^4}\right ) \, dx\\ &=\int \sqrt [4]{-x^3+x^5} \, dx-\int \frac {\sqrt [4]{-x^3+x^5}}{x^4} \, dx\\ &=\frac {4 \sqrt [4]{-x^3+x^5}}{9 x^3}+\frac {4}{9} x \sqrt [4]{-x^3+x^5}-\frac {2}{9} \int \frac {x}{\left (-x^3+x^5\right )^{3/4}} \, dx-\frac {2}{9} \int \frac {x^3}{\left (-x^3+x^5\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{-x^3+x^5}}{9 x^3}-\frac {8 \sqrt [4]{-x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{-x^3+x^5}+\frac {2}{9} \int \frac {x^3}{\left (-x^3+x^5\right )^{3/4}} \, dx-\frac {\left (2 x^{9/4} \left (-1+x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (-1+x^2\right )^{3/4}} \, dx}{9 \left (-x^3+x^5\right )^{3/4}}\\ &=\frac {4 \sqrt [4]{-x^3+x^5}}{9 x^3}-\frac {8 \sqrt [4]{-x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{-x^3+x^5}-\frac {\left (2 x^{9/4} \left (1-x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (1-x^2\right )^{3/4}} \, dx}{9 \left (-x^3+x^5\right )^{3/4}}+\frac {\left (2 x^{9/4} \left (-1+x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (-1+x^2\right )^{3/4}} \, dx}{9 \left (-x^3+x^5\right )^{3/4}}\\ &=\frac {4 \sqrt [4]{-x^3+x^5}}{9 x^3}-\frac {8 \sqrt [4]{-x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{-x^3+x^5}-\frac {8 x^4 \left (1-x^2\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};x^2\right )}{63 \left (-x^3+x^5\right )^{3/4}}+\frac {\left (2 x^{9/4} \left (1-x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (1-x^2\right )^{3/4}} \, dx}{9 \left (-x^3+x^5\right )^{3/4}}\\ &=\frac {4 \sqrt [4]{-x^3+x^5}}{9 x^3}-\frac {8 \sqrt [4]{-x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{-x^3+x^5}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 65, normalized size = 3.25 \begin {gather*} \frac {4 \sqrt [4]{x^3 \left (x^2-1\right )} \left (7 \, _2F_1\left (-\frac {9}{8},-\frac {1}{4};-\frac {1}{8};x^2\right )+9 x^4 \, _2F_1\left (-\frac {1}{4},\frac {7}{8};\frac {15}{8};x^2\right )\right )}{63 x^3 \sqrt [4]{1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 20, normalized size = 1.00 \begin {gather*} \frac {4 \left (-x^3+x^5\right )^{9/4}}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 26, normalized size = 1.30 \begin {gather*} \frac {4 \, {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 2 \, x^{2} + 1\right )}}{9 \, x^{3}} \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^{5} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 27, normalized size = 1.35
method | result | size |
trager | \(\frac {4 \left (x^{4}-2 x^{2}+1\right ) \left (x^{5}-x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(27\) |
gosper | \(\frac {4 \left (x^{5}-x^{3}\right )^{\frac {1}{4}} \left (x^{2}-1\right ) \left (-1+x \right ) \left (1+x \right )}{9 x^{3}}\) | \(28\) |
risch | \(\frac {4 \left (x^{3} \left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (x^{6}-3 x^{4}+3 x^{2}-1\right )}{9 x^{3} \left (x^{2}-1\right )}\) | \(39\) |
meijerg | \(\frac {4 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {9}{8}, -\frac {1}{4}\right ], \left [-\frac {1}{8}\right ], x^{2}\right )}{9 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}} x^{\frac {9}{4}}}+\frac {4 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {7}{8}\right ], \left [\frac {15}{8}\right ], x^{2}\right ) x^{\frac {7}{4}}}{7 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{4}}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 47, normalized size = 2.35 \begin {gather*} \frac {4\,{\left (x^5-x^3\right )}^{1/4}}{9\,x^3}-\frac {8\,{\left (x^5-x^3\right )}^{1/4}}{9\,x}+\frac {4\,x\,{\left (x^5-x^3\right )}^{1/4}}{9} \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 [4]{x^{3} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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