Optimal. Leaf size=28 \[ \frac {4 (2 x-1) \left (x^3-x^2\right )^{3/4}}{(x-1) x^2} \]
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Rubi [A] time = 0.13, antiderivative size = 36, normalized size of antiderivative = 1.29, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2056, 949, 74} \begin {gather*} \frac {4}{\sqrt [4]{x^3-x^2}}-\frac {8 (1-x)}{\sqrt [4]{x^3-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 949
Rule 2056
Rubi steps
\begin {align*} \int \frac {-2-x+2 x^2}{(-1+x) x \sqrt [4]{-x^2+x^3}} \, dx &=\frac {\left (\sqrt [4]{-1+x} \sqrt {x}\right ) \int \frac {-2-x+2 x^2}{(-1+x)^{5/4} x^{3/2}} \, dx}{\sqrt [4]{-x^2+x^3}}\\ &=\frac {4}{\sqrt [4]{-x^2+x^3}}-\frac {\left (4 \sqrt [4]{-1+x} \sqrt {x}\right ) \int \frac {-1-\frac {x}{2}}{\sqrt [4]{-1+x} x^{3/2}} \, dx}{\sqrt [4]{-x^2+x^3}}\\ &=\frac {4}{\sqrt [4]{-x^2+x^3}}-\frac {8 (1-x)}{\sqrt [4]{-x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 70, normalized size = 2.50 \begin {gather*} -\frac {4 \sqrt {x} \left (2 \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};1-x\right )-\, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};1-x\right )-2 \, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};1-x\right )\right )}{\sqrt [4]{(x-1) x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 28, normalized size = 1.00 \begin {gather*} \frac {4 (-1+2 x) \left (-x^2+x^3\right )^{3/4}}{(-1+x) x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 18, normalized size = 0.64 \begin {gather*} \frac {4 \, {\left (2 \, x - 1\right )}}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{4}}} \end {gather*}
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 \begin {gather*} \int \frac {2 \, x^{2} - x - 2}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{4}} {\left (x - 1\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 17, normalized size = 0.61
method | result | size |
risch | \(\frac {-4+8 x}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{4}}}\) | \(17\) |
gosper | \(\frac {-4+8 x}{\left (x^{3}-x^{2}\right )^{\frac {1}{4}}}\) | \(19\) |
trager | \(\frac {4 \left (-1+2 x \right ) \left (x^{3}-x^{2}\right )^{\frac {3}{4}}}{\left (-1+x \right ) x^{2}}\) | \(27\) |
meijerg | \(-\frac {4 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {1}{2}\right ], x\right )}{\mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \sqrt {x}}-\frac {4 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], x\right ) x^{\frac {3}{2}}}{3 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}}}+\frac {2 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], x\right ) \sqrt {x}}{\mathrm {signum}\left (-1+x \right )^{\frac {1}{4}}}\) | \(80\) |
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 \begin {gather*} \int \frac {2 \, x^{2} - x - 2}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{4}} {\left (x - 1\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 17, normalized size = 0.61 \begin {gather*} \frac {8\,x-4}{{\left (x^3-x^2\right )}^{1/4}} \end {gather*}
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 \begin {gather*} \int \frac {2 x^{2} - x - 2}{x \sqrt [4]{x^{2} \left (x - 1\right )} \left (x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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