Optimal. Leaf size=160 \[ \frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.24, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2139, 218, 2137, 203} \[ \frac {2 \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 218
Rule 2137
Rule 2139
Rubi steps
\begin {align*} \int \frac {x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx &=-\left (\frac {1}{3} \int \frac {1}{\sqrt {1-x^3}} \, dx\right )+\frac {1}{3} \int \frac {2^{2/3}+2 x}{\left (2^{2/3}-x\right ) \sqrt {1-x^3}} \, dx\\ &=\frac {2 \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 )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}}-\frac {1}{3} \left (2\ 2^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+3 x^2} \, dx,x,\frac {1-\sqrt [3]{2} x}{\sqrt {1-x^3}}\right )\\ &=-\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {1-x^3}}\right )}{3 \sqrt {3}}+\frac {2 \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 )}{3 \sqrt [4]{3} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}}\\ \end {align*}
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Mathematica [C] time = 0.33, size = 209, normalized size = 1.31 \[ \frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (-\frac {\left (x+\sqrt [3]{-1}\right ) \sqrt {\frac {(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {i 2^{2/3} \sqrt {x^2+x+1} \Pi \left (\frac {i \sqrt {3}}{\sqrt [3]{-1}+2^{2/3}};\sin ^{-1}\left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1}+2^{2/3}}\right )}{\sqrt {1-x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{3} + 2^{\frac {2}{3}} x^{2} + 2 \cdot 2^{\frac {1}{3}} x\right )} \sqrt {-x^{3} + 1}}{x^{6} - 5 \, x^{3} + 4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 253, normalized size = 1.58 \[ \frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}+\frac {2 i 2^{\frac {2}{3}} \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}-2^{\frac {2}{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x}{\sqrt {-x^{3} + 1} {\left (x - 2^{\frac {2}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x}{\sqrt {1-x^3}\,\left (x-2^{2/3}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x}{x \sqrt {1 - x^{3}} - 2^{\frac {2}{3}} \sqrt {1 - x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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