Optimal. Leaf size=137 \[ \frac{2}{5} x \sqrt{x^3-1}-\frac{4 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{5 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
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Rubi [A] time = 0.0203478, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {321, 219} \[ \frac{2}{5} x \sqrt{x^3-1}-\frac{4 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{5 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 219
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{-1+x^3}} \, dx &=\frac{2}{5} x \sqrt{-1+x^3}+\frac{2}{5} \int \frac{1}{\sqrt{-1+x^3}} \, dx\\ &=\frac{2}{5} x \sqrt{-1+x^3}-\frac{4 \sqrt{2-\sqrt{3}} (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 )}{5 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (1-\sqrt{3}-x\right )^2}} \sqrt{-1+x^3}}\\ \end{align*}
Mathematica [C] time = 0.0089929, size = 44, normalized size = 0.32 \[ \frac{2 x \left (\sqrt{1-x^3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};x^3\right )+x^3-1\right )}{5 \sqrt{x^3-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 127, normalized size = 0.9 \begin{align*}{\frac{2\,x}{5}\sqrt{{x}^{3}-1}}+{\frac{-6-2\,i\sqrt{3}}{5}\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 ) }}{\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 ){\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{x^{3}}{\sqrt{x^{3} - 1}}\,{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{x^{3}}{\sqrt{x^{3} - 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.795532, size = 27, normalized size = 0.2 \begin{align*} - \frac{i x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{7}{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{x^{3}}{\sqrt{x^{3} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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