Optimal. Leaf size=29 \[ \frac {3 (3-x) (1+x)}{4 \left (9+3 x-5 x^2+x^3\right )^{2/3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2092, 2089, 37}
\begin {gather*} \frac {3 (3-x) (x+1)}{4 \left (x^3-5 x^2+3 x+9\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 2089
Rule 2092
Rubi steps
\begin {align*} \int \frac {1}{\left (9+3 x-5 x^2+x^3\right )^{2/3}} \, dx &=\text {Subst}\left (\int \frac {1}{\left (\frac {128}{27}-\frac {16 x}{3}+x^3\right )^{2/3}} \, dx,x,-\frac {5}{3}+x\right )\\ &=\frac {\left (512 \sqrt [3]{2} (3-x)^{4/3} (1+x)^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {128}{9}-\frac {32 x}{3}\right )^{4/3} \left (\frac {128}{9}+\frac {16 x}{3}\right )^{2/3}} \, dx,x,-\frac {5}{3}+x\right )}{9 \left (9+3 x-5 x^2+x^3\right )^{2/3}}\\ &=\frac {3 (3-x) (1+x)}{4 \left (9+3 x-5 x^2+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.79 \begin {gather*} -\frac {3 (-3+x) (1+x)}{4 \left ((-3+x)^2 (1+x)\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 20, normalized size = 0.69
method | result | size |
risch | \(-\frac {3 \left (-3+x \right ) \left (1+x \right )}{4 \left (\left (1+x \right ) \left (-3+x \right )^{2}\right )^{\frac {2}{3}}}\) | \(20\) |
trager | \(-\frac {3 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {1}{3}}}{4 \left (-3+x \right )}\) | \(23\) |
gosper | \(-\frac {3 \left (1+x \right ) \left (-3+x \right )}{4 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {2}{3}}}\) | \(24\) |
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 = 0.38, size = 22, normalized size = 0.76 \begin {gather*} -\frac {3 \, {\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {1}{3}}}{4 \, {\left (x - 3\right )}} \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}{\left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac {2}{3}}}\, dx \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 24, normalized size = 0.83 \begin {gather*} -\frac {3\,{\left (x^3-5\,x^2+3\,x+9\right )}^{1/3}}{4\,\left (x-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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