Optimal. Leaf size=294 \[ \frac{5 (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}+\frac{5 \sqrt{x^3-1}}{8 \left (-x-\sqrt{3}+1\right )}+\frac{5 \sqrt{x^3-1}}{8 x}+\frac{\sqrt{x^3-1}}{4 x^4}-\frac{5 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16 \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
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Rubi [A] time = 0.0775788, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {325, 304, 219, 1879} \[ \frac{5 \sqrt{x^3-1}}{8 \left (-x-\sqrt{3}+1\right )}+\frac{5 \sqrt{x^3-1}}{8 x}+\frac{\sqrt{x^3-1}}{4 x^4}+\frac{5 (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{5 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{16 \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{1}{x^5 \sqrt{-1+x^3}} \, dx &=\frac{\sqrt{-1+x^3}}{4 x^4}+\frac{5}{8} \int \frac{1}{x^2 \sqrt{-1+x^3}} \, dx\\ &=\frac{\sqrt{-1+x^3}}{4 x^4}+\frac{5 \sqrt{-1+x^3}}{8 x}-\frac{5}{16} \int \frac{x}{\sqrt{-1+x^3}} \, dx\\ &=\frac{\sqrt{-1+x^3}}{4 x^4}+\frac{5 \sqrt{-1+x^3}}{8 x}+\frac{5}{16} \int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx-\frac{1}{8} \left (5 \sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )}\right ) \int \frac{1}{\sqrt{-1+x^3}} \, dx\\ &=\frac{5 \sqrt{-1+x^3}}{8 \left (1-\sqrt{3}-x\right )}+\frac{\sqrt{-1+x^3}}{4 x^4}+\frac{5 \sqrt{-1+x^3}}{8 x}-\frac{5 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{1+x+x^2}{\left (1-\sqrt{3}-x\right )^2}} E\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-x}{1-\sqrt{3}-x}\right )|-7+4 \sqrt{3}\right )}{16 \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}+\frac{5 (1-x) \sqrt{\frac{1+x+x^2}{\left (1-\sqrt{3}-x\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-x}{1-\sqrt{3}-x}\right )|-7+4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}\\ \end{align*}
Mathematica [C] time = 0.0056312, size = 40, normalized size = 0.14 \[ -\frac{\sqrt{1-x^3} \, _2F_1\left (-\frac{4}{3},\frac{1}{2};-\frac{1}{3};x^3\right )}{4 x^4 \sqrt{x^3-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 198, normalized size = 0.7 \begin{align*}{\frac{1}{4\,{x}^{4}}\sqrt{{x}^{3}-1}}+{\frac{5}{8\,x}\sqrt{{x}^{3}-1}}-{\frac{-{\frac{15}{2}}-{\frac{5\,i}{2}}\sqrt{3}}{8}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }} \left ( \left ({\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) + \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \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}{\sqrt{x^{3} - 1} 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{\sqrt{x^{3} - 1}}{x^{8} - x^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.05605, size = 34, normalized size = 0.12 \begin{align*} - \frac{i \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{2} \\ - \frac{1}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 x^{4} \Gamma \left (- \frac{1}{3}\right )} \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}{\sqrt{x^{3} - 1} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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