Optimal. Leaf size=23 \[ \frac {3 (3 x-2) \left (x^3+x^2\right )^{2/3}}{10 x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2016, 2014} \begin {gather*} \frac {9 \left (x^3+x^2\right )^{2/3}}{10 x^2}-\frac {3 \left (x^3+x^2\right )^{2/3}}{5 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt [3]{x^2+x^3}} \, dx &=-\frac {3 \left (x^2+x^3\right )^{2/3}}{5 x^3}-\frac {3}{5} \int \frac {1}{x \sqrt [3]{x^2+x^3}} \, dx\\ &=-\frac {3 \left (x^2+x^3\right )^{2/3}}{5 x^3}+\frac {9 \left (x^2+x^3\right )^{2/3}}{10 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {3 \left (x^2 (x+1)\right )^{2/3} (3 x-2)}{10 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 23, normalized size = 1.00 \begin {gather*} \frac {3 (-2+3 x) \left (x^2+x^3\right )^{2/3}}{10 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 19, normalized size = 0.83 \begin {gather*} \frac {3 \, {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}} {\left (3 \, x - 2\right )}}{10 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 19, normalized size = 0.83 \begin {gather*} -\frac {3}{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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maple [A] time = 0.05, size = 16, normalized size = 0.70
method | result | size |
meijerg | \(-\frac {3 \left (1-\frac {3 x}{2}\right ) \left (1+x \right )^{\frac {2}{3}}}{5 x^{\frac {5}{3}}}\) | \(16\) |
trager | \(\frac {3 \left (-2+3 x \right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}}{10 x^{3}}\) | \(20\) |
gosper | \(\frac {3 \left (1+x \right ) \left (-2+3 x \right )}{10 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}}\) | \(23\) |
risch | \(\frac {-\frac {3}{5}+\frac {3}{10} x +\frac {9}{10} x^{2}}{x \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}\) | \(23\) |
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 {1}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 29, normalized size = 1.26 \begin {gather*} \frac {9\,x\,{\left (x^3+x^2\right )}^{2/3}-6\,{\left (x^3+x^2\right )}^{2/3}}{10\,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 {1}{x^{2} \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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