Integrand size = 15, antiderivative size = 38 \[ \int \sqrt [3]{1-x} (1+x)^2 \, dx=-3 (1-x)^{4/3}+\frac {12}{7} (1-x)^{7/3}-\frac {3}{10} (1-x)^{10/3} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \sqrt [3]{1-x} (1+x)^2 \, dx=-\frac {3}{10} (1-x)^{10/3}+\frac {12}{7} (1-x)^{7/3}-3 (1-x)^{4/3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (4 \sqrt [3]{1-x}-4 (1-x)^{4/3}+(1-x)^{7/3}\right ) \, dx \\ & = -3 (1-x)^{4/3}+\frac {12}{7} (1-x)^{7/3}-\frac {3}{10} (1-x)^{10/3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int \sqrt [3]{1-x} (1+x)^2 \, dx=-\frac {3}{70} (1-x)^{4/3} \left (37+26 x+7 x^2\right ) \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53
method | result | size |
gosper | \(-\frac {3 \left (1-x \right )^{\frac {4}{3}} \left (7 x^{2}+26 x +37\right )}{70}\) | \(20\) |
trager | \(\left (\frac {3}{10} x^{3}+\frac {57}{70} x^{2}+\frac {33}{70} x -\frac {111}{70}\right ) \left (1-x \right )^{\frac {1}{3}}\) | \(24\) |
pseudoelliptic | \(\frac {3 \left (1-x \right )^{\frac {1}{3}} \left (7 x^{3}+19 x^{2}+11 x -37\right )}{70}\) | \(25\) |
risch | \(-\frac {3 \left (7 x^{3}+19 x^{2}+11 x -37\right ) \left (-1+x \right )}{70 \left (1-x \right )^{\frac {2}{3}}}\) | \(28\) |
derivativedivides | \(-3 \left (1-x \right )^{\frac {4}{3}}+\frac {12 \left (1-x \right )^{\frac {7}{3}}}{7}-\frac {3 \left (1-x \right )^{\frac {10}{3}}}{10}\) | \(29\) |
default | \(-3 \left (1-x \right )^{\frac {4}{3}}+\frac {12 \left (1-x \right )^{\frac {7}{3}}}{7}-\frac {3 \left (1-x \right )^{\frac {10}{3}}}{10}\) | \(29\) |
meijerg | \(x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},1;2;x \right )+x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},2;3;x \right )+\frac {x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},3;4;x \right )}{3}\) | \(34\) |
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Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63 \[ \int \sqrt [3]{1-x} (1+x)^2 \, dx=\frac {3}{70} \, {\left (7 \, x^{3} + 19 \, x^{2} + 11 \, x - 37\right )} {\left (-x + 1\right )}^{\frac {1}{3}} \]
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Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 3.79 \[ \int \sqrt [3]{1-x} (1+x)^2 \, dx=\begin {cases} - \frac {3 \sqrt [3]{x - 1} \left (x + 1\right )^{3} e^{- \frac {2 i \pi }{3}}}{10} + \frac {3 \sqrt [3]{x - 1} \left (x + 1\right )^{2} e^{- \frac {2 i \pi }{3}}}{35} + \frac {9 \sqrt [3]{x - 1} \left (x + 1\right ) e^{- \frac {2 i \pi }{3}}}{35} + \frac {54 \sqrt [3]{x - 1} e^{- \frac {2 i \pi }{3}}}{35} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {3 \sqrt [3]{1 - x} \left (x + 1\right )^{3}}{10} - \frac {3 \sqrt [3]{1 - x} \left (x + 1\right )^{2}}{35} - \frac {9 \sqrt [3]{1 - x} \left (x + 1\right )}{35} - \frac {54 \sqrt [3]{1 - x}}{35} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \sqrt [3]{1-x} (1+x)^2 \, dx=-\frac {3}{10} \, {\left (-x + 1\right )}^{\frac {10}{3}} + \frac {12}{7} \, {\left (-x + 1\right )}^{\frac {7}{3}} - 3 \, {\left (-x + 1\right )}^{\frac {4}{3}} \]
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Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \sqrt [3]{1-x} (1+x)^2 \, dx=\frac {3}{10} \, {\left (x - 1\right )}^{3} {\left (-x + 1\right )}^{\frac {1}{3}} + \frac {12}{7} \, {\left (x - 1\right )}^{2} {\left (-x + 1\right )}^{\frac {1}{3}} - 3 \, {\left (-x + 1\right )}^{\frac {4}{3}} \]
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Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.55 \[ \int \sqrt [3]{1-x} (1+x)^2 \, dx=-\frac {3\,{\left (1-x\right )}^{4/3}\,\left (40\,x+7\,{\left (x-1\right )}^2+30\right )}{70} \]
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