Integrand size = 9, antiderivative size = 23 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{4} (-1+z)^{4/3}+\frac {3}{7} (-1+z)^{7/3} \]
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Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{7} (z-1)^{7/3}+\frac {3}{4} (z-1)^{4/3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt [3]{-1+z}+(-1+z)^{4/3}\right ) \, dz \\ & = \frac {3}{4} (-1+z)^{4/3}+\frac {3}{7} (-1+z)^{7/3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{28} (7+4 (-1+z)) (-1+z)^{4/3} \]
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Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(\frac {3 \left (-1+z \right )^{\frac {4}{3}} \left (4 z +3\right )}{28}\) | \(13\) |
derivativedivides | \(\frac {3 \left (-1+z \right )^{\frac {4}{3}}}{4}+\frac {3 \left (-1+z \right )^{\frac {7}{3}}}{7}\) | \(16\) |
default | \(\frac {3 \left (-1+z \right )^{\frac {4}{3}}}{4}+\frac {3 \left (-1+z \right )^{\frac {7}{3}}}{7}\) | \(16\) |
trager | \(\left (\frac {3}{7} z^{2}-\frac {3}{28} z -\frac {9}{28}\right ) \left (-1+z \right )^{\frac {1}{3}}\) | \(17\) |
risch | \(\frac {3 \left (-1+z \right )^{\frac {1}{3}} \left (4 z^{2}-z -3\right )}{28}\) | \(18\) |
meijerg | \(\frac {\operatorname {signum}\left (-1+z \right )^{\frac {1}{3}} z^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},2;3;z \right )}{2 \left (-\operatorname {signum}\left (-1+z \right )\right )^{\frac {1}{3}}}\) | \(27\) |
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none
Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{28} \, {\left (4 \, z^{2} - z - 3\right )} {\left (z - 1\right )}^{\frac {1}{3}} \]
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Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00 \[ \int \sqrt [3]{-1+z} z \, dz=\begin {cases} \frac {3 z^{2} \sqrt [3]{z - 1}}{7} - \frac {3 z \sqrt [3]{z - 1}}{28} - \frac {9 \sqrt [3]{z - 1}}{28} & \text {for}\: \left |{z}\right | > 1 \\\frac {3 z^{2} \sqrt [3]{1 - z} e^{\frac {i \pi }{3}}}{7} - \frac {3 z \sqrt [3]{1 - z} e^{\frac {i \pi }{3}}}{28} - \frac {9 \sqrt [3]{1 - z} e^{\frac {i \pi }{3}}}{28} & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{7} \, {\left (z - 1\right )}^{\frac {7}{3}} + \frac {3}{4} \, {\left (z - 1\right )}^{\frac {4}{3}} \]
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none
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3}{7} \, {\left (z - 1\right )}^{\frac {7}{3}} + \frac {3}{4} \, {\left (z - 1\right )}^{\frac {4}{3}} \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \sqrt [3]{-1+z} z \, dz=\frac {3\,\left (4\,z+3\right )\,{\left (z-1\right )}^{4/3}}{28} \]
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