Optimal. Leaf size=264 \[ \frac{8 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{7 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{2}{7} \sqrt{-x^3-1} x^2+\frac{8 \sqrt{-x^3-1}}{7 \left (x-\sqrt{3}+1\right )}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]
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Rubi [A] time = 0.0543119, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {321, 304, 219, 1879} \[ -\frac{2}{7} \sqrt{-x^3-1} x^2+\frac{8 \sqrt{-x^3-1}}{7 \left (x-\sqrt{3}+1\right )}+\frac{8 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{4 \sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{7 \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{x^4}{\sqrt{-1-x^3}} \, dx &=-\frac{2}{7} x^2 \sqrt{-1-x^3}-\frac{4}{7} \int \frac{x}{\sqrt{-1-x^3}} \, dx\\ &=-\frac{2}{7} x^2 \sqrt{-1-x^3}-\frac{4}{7} \int \frac{1+\sqrt{3}+x}{\sqrt{-1-x^3}} \, dx+\frac{1}{7} \left (4 \sqrt{2 \left (2+\sqrt{3}\right )}\right ) \int \frac{1}{\sqrt{-1-x^3}} \, dx\\ &=-\frac{2}{7} x^2 \sqrt{-1-x^3}+\frac{8 \sqrt{-1-x^3}}{7 \left (1-\sqrt{3}+x\right )}-\frac{4 \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 )}{7 \sqrt{-\frac{1+x}{\left (1-\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}+\frac{8 \sqrt{2} (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 )}{7 \sqrt [4]{3} \sqrt{-\frac{1+x}{\left (1-\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}\\ \end{align*}
Mathematica [C] time = 0.0099205, size = 49, normalized size = 0.19 \[ \frac{2 x^2 \left (-\sqrt{x^3+1} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right )+x^3+1\right )}{7 \sqrt{-x^3-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 175, normalized size = 0.7 \begin{align*} -{\frac{2\,{x}^{2}}{7}\sqrt{-{x}^{3}-1}}+{{\frac{8\,i}{21}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}} \left ( \left ({\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) -{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\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{x^{4}}{\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{\sqrt{-x^{3} - 1} x^{4}}{x^{3} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.79921, size = 32, normalized size = 0.12 \begin{align*} - \frac{i x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{8}{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^{4}}{\sqrt{-x^{3} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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