Optimal. Leaf size=308 \[ -\frac{1}{2} x^2 \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};x^3\right )-\frac{\left (1-x^3\right )^{2/3}}{x}-\frac{3 \left (1-x^3\right )^{2/3}}{4 x^4}+\frac{1}{2 \sqrt [3]{1-x^3} x^4}+\frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{12 \sqrt [3]{2}}-\frac{\log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{8 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{4 \sqrt [3]{2} \sqrt{3}}+\frac{\log \left ((1-x) (x+1)^2\right )}{24 \sqrt [3]{2}} \]
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Rubi [C] time = 0.0196007, antiderivative size = 26, normalized size of antiderivative = 0.08, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {510} \[ -\frac{F_1\left (-\frac{4}{3};\frac{4}{3},1;-\frac{1}{3};x^3,-x^3\right )}{4 x^4} \]
Warning: Unable to verify antiderivative.
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Rule 510
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=-\frac{F_1\left (-\frac{4}{3};\frac{4}{3},1;-\frac{1}{3};x^3,-x^3\right )}{4 x^4}\\ \end{align*}
Mathematica [C] time = 0.0811564, size = 79, normalized size = 0.26 \[ -\frac{4 x^9 F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};x^3,-x^3\right )+5 x^6 F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};x^3,-x^3\right )+\frac{5 \left (-4 x^6+x^3+1\right )}{\sqrt [3]{1-x^3}}}{20 x^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({x}^{3}+1 \right ){x}^{5}} \left ( -{x}^{3}+1 \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{4}{3}} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{14} - x^{11} - x^{8} + x^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{4}{3}} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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