Optimal. Leaf size=20 \[ \frac {4 \left (x^4-x^3\right )^{9/4}}{9 x^9} \]
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Rubi [B] time = 0.13, antiderivative size = 41, normalized size of antiderivative = 2.05, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2052, 2016, 2014} \begin {gather*} \frac {4 \left (x^4-x^3\right )^{5/4}}{9 x^5}-\frac {4 \left (x^4-x^3\right )^{5/4}}{9 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rule 2052
Rubi steps
\begin {align*} \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x^4} \, dx &=\int \left (-\frac {\sqrt [4]{-x^3+x^4}}{x^4}+\frac {\sqrt [4]{-x^3+x^4}}{x^3}\right ) \, dx\\ &=-\int \frac {\sqrt [4]{-x^3+x^4}}{x^4} \, dx+\int \frac {\sqrt [4]{-x^3+x^4}}{x^3} \, dx\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}+\frac {4 \left (-x^3+x^4\right )^{5/4}}{5 x^5}-\frac {4}{9} \int \frac {\sqrt [4]{-x^3+x^4}}{x^3} \, dx\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}+\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} \frac {4 \left ((x-1) x^3\right )^{9/4}}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 20, normalized size = 1.00 \begin {gather*} \frac {4 \left (-x^3+x^4\right )^{9/4}}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 24, normalized size = 1.20 \begin {gather*} \frac {4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 2 \, x + 1\right )}}{9 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 18, normalized size = 0.90 \begin {gather*} -\frac {4}{9} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 22, normalized size = 1.10
method | result | size |
gosper | \(\frac {4 \left (-1+x \right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(22\) |
trager | \(\frac {4 \left (x^{2}-2 x +1\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(25\) |
risch | \(\frac {4 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x^{3}-3 x^{2}+3 x -1\right )}{9 \left (-1+x \right ) x^{3}}\) | \(33\) |
meijerg | \(\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \left (-\frac {4}{5} x^{2}-\frac {1}{5} x +1\right ) \left (1-x \right )^{\frac {1}{4}}}{9 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {9}{4}}}-\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \left (1-x \right )^{\frac {5}{4}}}{5 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {5}{4}}}\) | \(64\) |
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^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x - 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 = 0.25, size = 49, normalized size = 2.45 \begin {gather*} \frac {4\,x^2\,{\left (x^4-x^3\right )}^{1/4}-8\,x\,{\left (x^4-x^3\right )}^{1/4}+4\,{\left (x^4-x^3\right )}^{1/4}}{9\,x^3} \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 )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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