Integrand size = 39, antiderivative size = 46 \[ \int \frac {-2 x^5+3 x^8-x^2 \left (-1+3 x^3\right )^{2/3}}{\left (-1+3 x^3\right )^{3/4}} \, dx=-\frac {4}{27} \sqrt [4]{-1+3 x^3}-\frac {4}{33} \left (-1+3 x^3\right )^{11/12}+\frac {4}{243} \left (-1+3 x^3\right )^{9/4} \]
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Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6874, 272, 45, 267} \[ \int \frac {-2 x^5+3 x^8-x^2 \left (-1+3 x^3\right )^{2/3}}{\left (-1+3 x^3\right )^{3/4}} \, dx=\frac {4}{243} \left (3 x^3-1\right )^{9/4}-\frac {4}{33} \left (3 x^3-1\right )^{11/12}-\frac {4}{27} \sqrt [4]{3 x^3-1} \]
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Rule 45
Rule 267
Rule 272
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 x^5}{\left (-1+3 x^3\right )^{3/4}}+\frac {3 x^8}{\left (-1+3 x^3\right )^{3/4}}-\frac {x^2}{\sqrt [12]{-1+3 x^3}}\right ) \, dx \\ & = -\left (2 \int \frac {x^5}{\left (-1+3 x^3\right )^{3/4}} \, dx\right )+3 \int \frac {x^8}{\left (-1+3 x^3\right )^{3/4}} \, dx-\int \frac {x^2}{\sqrt [12]{-1+3 x^3}} \, dx \\ & = -\frac {4}{33} \left (-1+3 x^3\right )^{11/12}-\frac {2}{3} \text {Subst}\left (\int \frac {x}{(-1+3 x)^{3/4}} \, dx,x,x^3\right )+\text {Subst}\left (\int \frac {x^2}{(-1+3 x)^{3/4}} \, dx,x,x^3\right ) \\ & = -\frac {4}{33} \left (-1+3 x^3\right )^{11/12}-\frac {2}{3} \text {Subst}\left (\int \left (\frac {1}{3 (-1+3 x)^{3/4}}+\frac {1}{3} \sqrt [4]{-1+3 x}\right ) \, dx,x,x^3\right )+\text {Subst}\left (\int \left (\frac {1}{9 (-1+3 x)^{3/4}}+\frac {2}{9} \sqrt [4]{-1+3 x}+\frac {1}{9} (-1+3 x)^{5/4}\right ) \, dx,x,x^3\right ) \\ & = -\frac {4}{27} \sqrt [4]{-1+3 x^3}-\frac {4}{33} \left (-1+3 x^3\right )^{11/12}+\frac {4}{243} \left (-1+3 x^3\right )^{9/4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.87 \[ \int \frac {-2 x^5+3 x^8-x^2 \left (-1+3 x^3\right )^{2/3}}{\left (-1+3 x^3\right )^{3/4}} \, dx=-\frac {4 \sqrt [4]{-1+3 x^3} \left (88+66 x^3-99 x^6+81 \left (-1+3 x^3\right )^{2/3}\right )}{2673} \]
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Result contains higher order function than in optimal. Order 9 vs. order 2.
Time = 0.36 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.52
method | result | size |
meijerg | \(-\frac {{\left (-\operatorname {signum}\left (3 x^{3}-1\right )\right )}^{\frac {3}{4}} x^{6} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{4},2;3;3 x^{3}\right )}{3 \operatorname {signum}\left (3 x^{3}-1\right )^{\frac {3}{4}}}+\frac {{\left (-\operatorname {signum}\left (3 x^{3}-1\right )\right )}^{\frac {3}{4}} x^{9} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{4},3;4;3 x^{3}\right )}{3 \operatorname {signum}\left (3 x^{3}-1\right )^{\frac {3}{4}}}-\frac {{\left (-\operatorname {signum}\left (3 x^{3}-1\right )\right )}^{\frac {1}{12}} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{12},1;2;3 x^{3}\right )}{3 \operatorname {signum}\left (3 x^{3}-1\right )^{\frac {1}{12}}}\) | \(116\) |
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Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \frac {-2 x^5+3 x^8-x^2 \left (-1+3 x^3\right )^{2/3}}{\left (-1+3 x^3\right )^{3/4}} \, dx=\frac {4}{243} \, {\left (9 \, x^{6} - 6 \, x^{3} - 8\right )} {\left (3 \, x^{3} - 1\right )}^{\frac {1}{4}} - \frac {4}{33} \, {\left (3 \, x^{3} - 1\right )}^{\frac {11}{12}} \]
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Result contains complex when optimal does not.
Time = 4.67 (sec) , antiderivative size = 221, normalized size of antiderivative = 4.80 \[ \int \frac {-2 x^5+3 x^8-x^2 \left (-1+3 x^3\right )^{2/3}}{\left (-1+3 x^3\right )^{3/4}} \, dx=- \frac {4 \left (3 x^{3} - 1\right )^{\frac {11}{12}}}{33} - 2 \left (\begin {cases} \frac {4 x^{3} \sqrt [4]{3 x^{3} - 1}}{45} + \frac {16 \sqrt [4]{3 x^{3} - 1}}{135} & \text {for}\: \left |{x^{3}}\right | > \frac {1}{3} \\- \frac {4 x^{3} \sqrt [4]{1 - 3 x^{3}} e^{- \frac {3 i \pi }{4}}}{45} - \frac {16 \sqrt [4]{1 - 3 x^{3}} e^{- \frac {3 i \pi }{4}}}{135} & \text {otherwise} \end {cases}\right ) + 3 \left (\begin {cases} \frac {4 x^{6} \sqrt [4]{3 x^{3} - 1}}{81} + \frac {32 x^{3} \sqrt [4]{3 x^{3} - 1}}{1215} + \frac {128 \sqrt [4]{3 x^{3} - 1}}{3645} & \text {for}\: \left |{x^{3}}\right | > \frac {1}{3} \\\frac {4 x^{6} \sqrt [4]{1 - 3 x^{3}} e^{\frac {i \pi }{4}}}{81} + \frac {32 x^{3} \sqrt [4]{1 - 3 x^{3}} e^{\frac {i \pi }{4}}}{1215} + \frac {128 \sqrt [4]{1 - 3 x^{3}} e^{\frac {i \pi }{4}}}{3645} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \frac {-2 x^5+3 x^8-x^2 \left (-1+3 x^3\right )^{2/3}}{\left (-1+3 x^3\right )^{3/4}} \, dx=\frac {4}{243} \, {\left (3 \, x^{3} - 1\right )}^{\frac {9}{4}} - \frac {4}{33} \, {\left (3 \, x^{3} - 1\right )}^{\frac {11}{12}} - \frac {4}{27} \, {\left (3 \, x^{3} - 1\right )}^{\frac {1}{4}} \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \frac {-2 x^5+3 x^8-x^2 \left (-1+3 x^3\right )^{2/3}}{\left (-1+3 x^3\right )^{3/4}} \, dx=\frac {4}{243} \, {\left (3 \, x^{3} - 1\right )}^{\frac {9}{4}} - \frac {4}{33} \, {\left (3 \, x^{3} - 1\right )}^{\frac {11}{12}} - \frac {4}{27} \, {\left (3 \, x^{3} - 1\right )}^{\frac {1}{4}} \]
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Time = 0.45 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \frac {-2 x^5+3 x^8-x^2 \left (-1+3 x^3\right )^{2/3}}{\left (-1+3 x^3\right )^{3/4}} \, dx=-{\left (3\,x^3-1\right )}^{1/4}\,\left (\frac {8\,x^3}{81}-\frac {4\,x^6}{27}+\frac {4\,{\left (3\,x^3-1\right )}^{2/3}}{33}+\frac {32}{243}\right ) \]
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