Optimal. Leaf size=30 \[ \frac {3 \left (5+6 x+9 x^2\right ) \left (-x^2+x^3\right )^{2/3}}{40 x^4} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(30)=60\).
time = 0.05, antiderivative size = 61, normalized size of antiderivative = 2.03, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039}
\begin {gather*} \frac {27 \left (x^3-x^2\right )^{2/3}}{40 x^2}+\frac {9 \left (x^3-x^2\right )^{2/3}}{20 x^3}+\frac {3 \left (x^3-x^2\right )^{2/3}}{8 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt [3]{-x^2+x^3}} \, dx &=\frac {3 \left (-x^2+x^3\right )^{2/3}}{8 x^4}+\frac {3}{4} \int \frac {1}{x^2 \sqrt [3]{-x^2+x^3}} \, dx\\ &=\frac {3 \left (-x^2+x^3\right )^{2/3}}{8 x^4}+\frac {9 \left (-x^2+x^3\right )^{2/3}}{20 x^3}+\frac {9}{20} \int \frac {1}{x \sqrt [3]{-x^2+x^3}} \, dx\\ &=\frac {3 \left (-x^2+x^3\right )^{2/3}}{8 x^4}+\frac {9 \left (-x^2+x^3\right )^{2/3}}{20 x^3}+\frac {27 \left (-x^2+x^3\right )^{2/3}}{40 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 28, normalized size = 0.93 \begin {gather*} \frac {3 \left ((-1+x) x^2\right )^{2/3} \left (5+6 x+9 x^2\right )}{40 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 27, normalized size = 0.90
method | result | size |
trager | \(\frac {3 \left (9 x^{2}+6 x +5\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}}{40 x^{4}}\) | \(27\) |
gosper | \(\frac {3 \left (-1+x \right ) \left (9 x^{2}+6 x +5\right )}{40 x^{2} \left (x^{3}-x^{2}\right )^{\frac {1}{3}}}\) | \(30\) |
risch | \(\frac {-\frac {3}{8}-\frac {3}{40} x -\frac {9}{40} x^{2}+\frac {27}{40} x^{3}}{x^{2} \left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}\) | \(30\) |
meijerg | \(-\frac {3 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} \left (\frac {9}{5} x^{2}+\frac {6}{5} x +1\right ) \left (1-x \right )^{\frac {2}{3}}}{8 \mathrm {signum}\left (-1+x \right )^{\frac {1}{3}} x^{\frac {8}{3}}}\) | \(37\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 26, normalized size = 0.87 \begin {gather*} \frac {3 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}} {\left (9 \, x^{2} + 6 \, x + 5\right )}}{40 \, x^{4}} \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 {1}{x^{3} \sqrt [3]{x^{2} \left (x - 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 41, normalized size = 1.37 \begin {gather*} \frac {3}{8} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} - \frac {6}{5} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{3}} + \frac {3}{2} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 49, normalized size = 1.63 \begin {gather*} \frac {27\,x^2\,{\left (x^3-x^2\right )}^{2/3}+18\,x\,{\left (x^3-x^2\right )}^{2/3}+15\,{\left (x^3-x^2\right )}^{2/3}}{40\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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