Optimal. Leaf size=76 \[ \frac {\left (210 x^4-7 x^3-3 x^2+681 x-229\right ) \sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}}{910 (x+1) \left (x^2-x+1\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 139, normalized size of antiderivative = 1.83, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6688, 6719, 1850} \begin {gather*} \frac {\sqrt [3]{(1-3 x) \left (x^3+1\right )^3} (1-3 x)^4}{351 \left (x^3+1\right )}-\frac {\sqrt [3]{(1-3 x) \left (x^3+1\right )^3} (1-3 x)^3}{90 \left (x^3+1\right )}+\frac {\sqrt [3]{(1-3 x) \left (x^3+1\right )^3} (1-3 x)^2}{63 \left (x^3+1\right )}-\frac {7 \sqrt [3]{(1-3 x) \left (x^3+1\right )^3} (1-3 x)}{27 \left (x^3+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 1850
Rule 6688
Rule 6719
Rubi steps
\begin {align*} \int \sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}} \, dx &=\int \sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \, dx\\ &=\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \sqrt [3]{-1+3 x} \left (1+x^3\right ) \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}\\ &=\frac {\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )} \int \left (\frac {28}{27} \sqrt [3]{-1+3 x}+\frac {1}{9} (-1+3 x)^{4/3}+\frac {1}{9} (-1+3 x)^{7/3}+\frac {1}{27} (-1+3 x)^{10/3}\right ) \, dx}{\sqrt [3]{-1+3 x} \left (1+x^3\right )}\\ &=-\frac {7 (1-3 x) \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}{27 \left (1+x^3\right )}+\frac {(1-3 x)^2 \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}{63 \left (1+x^3\right )}-\frac {(1-3 x)^3 \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}{90 \left (1+x^3\right )}+\frac {(1-3 x)^4 \sqrt [3]{(1-3 x) \left (1+x^3\right )^3}}{351 \left (1+x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.58 \begin {gather*} -\frac {\left (-\left ((3 x-1) \left (x^3+1\right )^3\right )\right )^{4/3} \left (70 x^3+21 x^2+6 x+229\right )}{910 \left (x^3+1\right )^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 71, normalized size = 0.93 \begin {gather*} \frac {\left (-229-6 x-21 x^2-70 x^3\right ) \left (1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}\right )^{4/3}}{910 (1+x)^4 \left (1-x+x^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 64, normalized size = 0.84 \begin {gather*} \frac {{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}} {\left (210 \, x^{4} - 7 \, x^{3} - 3 \, x^{2} + 681 \, x - 229\right )}}{910 \, {\left (x^{3} + 1\right )}} \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 {\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 46, normalized size = 0.61
method | result | size |
risch | \(\frac {\left (-\left (-1+3 x \right ) \left (x^{3}+1\right )^{3}\right )^{\frac {1}{3}} \left (210 x^{4}-7 x^{3}-3 x^{2}+681 x -229\right )}{910 x^{3}+910}\) | \(46\) |
trager | \(\frac {\left (210 x^{4}-7 x^{3}-3 x^{2}+681 x -229\right ) \left (-3 x^{10}+x^{9}-9 x^{7}+3 x^{6}-9 x^{4}+3 x^{3}-3 x +1\right )^{\frac {1}{3}}}{910 x^{3}+910}\) | \(65\) |
gosper | \(\frac {\left (-1+3 x \right ) \left (70 x^{3}+21 x^{2}+6 x +229\right ) \left (-3 x^{10}+x^{9}-9 x^{7}+3 x^{6}-9 x^{4}+3 x^{3}-3 x +1\right )^{\frac {1}{3}}}{910 \left (1+x \right ) \left (x^{2}-x +1\right )}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 29, normalized size = 0.38 \begin {gather*} -\frac {1}{910} \, {\left (210 \, x^{4} - 7 \, x^{3} - 3 \, x^{2} + 681 \, x - 229\right )} {\left (3 \, x - 1\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 64, normalized size = 0.84 \begin {gather*} -\frac {\left (-\frac {3\,x^4}{13}+\frac {x^3}{130}+\frac {3\,x^2}{910}-\frac {681\,x}{910}+\frac {229}{910}\right )\,{\left (-3\,x^{10}+x^9-9\,x^7+3\,x^6-9\,x^4+3\,x^3-3\,x+1\right )}^{1/3}}{x^3+1} \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 \sqrt [3]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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