Optimal. Leaf size=46 \[ \frac{4}{243} \left (3 x^3-1\right )^{9/4}-\frac{4}{33} \left (3 x^3-1\right )^{11/12}-\frac{4}{27} \sqrt [4]{3 x^3-1} \]
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Rubi [A] time = 0.193269, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {6742, 266, 43, 261} \[ \frac{4}{243} \left (3 x^3-1\right )^{9/4}-\frac{4}{33} \left (3 x^3-1\right )^{11/12}-\frac{4}{27} \sqrt [4]{3 x^3-1} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 266
Rule 43
Rule 261
Rubi steps
\begin{align*} \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 &=\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} \operatorname{Subst}\left (\int \frac{x}{(-1+3 x)^{3/4}} \, dx,x,x^3\right )+\operatorname{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} \operatorname{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 )+\operatorname{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*}
Mathematica [A] time = 0.0805153, size = 40, normalized size = 0.87 \[ -\frac{4 \sqrt [4]{3 x^3-1} \left (-99 x^6+66 x^3+81 \left (3 x^3-1\right )^{2/3}+88\right )}{2673} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.067, size = 116, normalized size = 2.5 \begin{align*} -{\frac{{x}^{6}}{3} \left ( -{\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{{\frac{3}{4}}}{\mbox{$_2$F$_1$}({\frac{3}{4}},2;\,3;\,3\,{x}^{3})} \left ({\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{-{\frac{3}{4}}}}+{\frac{{x}^{9}}{3} \left ( -{\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{{\frac{3}{4}}}{\mbox{$_2$F$_1$}({\frac{3}{4}},3;\,4;\,3\,{x}^{3})} \left ({\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{-{\frac{3}{4}}}}-{\frac{{x}^{3}}{3} \left ( -{\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{{\frac{1}{12}}}{\mbox{$_2$F$_1$}({\frac{1}{12}},1;\,2;\,3\,{x}^{3})} \left ({\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{-{\frac{1}{12}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972361, size = 46, normalized size = 1. \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03497, size = 97, normalized size = 2.11 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.08764, size = 221, normalized size = 4.8 \begin{align*} - \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}\: 3 \left |{x^{3}}\right | > 1 \\- \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}\: 3 \left |{x^{3}}\right | > 1 \\\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07351, size = 46, normalized size = 1. \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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