Optimal. Leaf size=39 \[ x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {(1-2 x) x+1}{\sqrt [3]{1-x^3}} \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1854, 12, 364} \[ x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {(1-2 x) x+1}{\sqrt [3]{1-x^3}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 364
Rule 1854
Rubi steps
\begin {align*} \int \frac {(1-x)^2}{\left (1-x^3\right )^{4/3}} \, dx &=\frac {1+(1-2 x) x}{\sqrt [3]{1-x^3}}-\int -\frac {2 x}{\sqrt [3]{1-x^3}} \, dx\\ &=\frac {1+(1-2 x) x}{\sqrt [3]{1-x^3}}+2 \int \frac {x}{\sqrt [3]{1-x^3}} \, dx\\ &=\frac {1+(1-2 x) x}{\sqrt [3]{1-x^3}}+x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 1.10 \[ x^2 \left (-\, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {5}{3};x^3\right )\right )+\frac {x}{\sqrt [3]{1-x^3}}+\frac {1}{\sqrt [3]{1-x^3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x - 1\right )}^{2}}{{\left (-x^{3} + 1\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 34, normalized size = 0.87 \[ x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )-\frac {\left (x -1\right ) \left (2 x +1\right )}{\left (-x^{3}+1\right )^{\frac {1}{3}}} \]
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 \[ \frac {x}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}}} - \int \frac {x^{2} - 2 \, x}{{\left (x^{3} - 1\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (x-1\right )}^2}{{\left (1-x^3\right )}^{4/3}} \,d x \]
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 \[ \int \frac {\left (x - 1\right )^{2}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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