Optimal. Leaf size=25 \[ x+\left (3-5 \left (4+\left (x-x^2\right )^{\sqrt [4]{e}}\right )^2\right )^2 \]
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Rubi [B] time = 1.91, antiderivative size = 68, normalized size of antiderivative = 2.72, number of steps used = 11, number of rules used = 4, integrand size = 119, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {1593, 6742, 6688, 629} \begin {gather*} 25 \left (x-x^2\right )^{4 \sqrt [4]{e}}+400 \left (x-x^2\right )^{3 \sqrt [4]{e}}+2370 \left (x-x^2\right )^{2 \sqrt [4]{e}}+6160 \left (x-x^2\right )^{\sqrt [4]{e}}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rule 1593
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x+x^2+\sqrt [4]{e} (-6160+12320 x) \left (x-x^2\right )^{\sqrt [4]{e}}+\sqrt [4]{e} (-4740+9480 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}+\sqrt [4]{e} (-1200+2400 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}+\sqrt [4]{e} (-100+200 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{(-1+x) x} \, dx\\ &=\int \left (1+\frac {6160 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{\sqrt [4]{e}}}{(1-x) x}+\frac {4740 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}}{(1-x) x}+\frac {1200 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}}{(1-x) x}+\frac {100 \sqrt [4]{e} (1-2 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{(1-x) x}\right ) \, dx\\ &=x+\left (100 \sqrt [4]{e}\right ) \int \frac {(1-2 x) \left (x-x^2\right )^{4 \sqrt [4]{e}}}{(1-x) x} \, dx+\left (1200 \sqrt [4]{e}\right ) \int \frac {(1-2 x) \left (x-x^2\right )^{3 \sqrt [4]{e}}}{(1-x) x} \, dx+\left (4740 \sqrt [4]{e}\right ) \int \frac {(1-2 x) \left (x-x^2\right )^{2 \sqrt [4]{e}}}{(1-x) x} \, dx+\left (6160 \sqrt [4]{e}\right ) \int \frac {(1-2 x) \left (x-x^2\right )^{\sqrt [4]{e}}}{(1-x) x} \, dx\\ &=x+\left (100 \sqrt [4]{e}\right ) \int (1-2 x) \left (x-x^2\right )^{-1+4 \sqrt [4]{e}} \, dx+\left (1200 \sqrt [4]{e}\right ) \int (1-2 x) \left (x-x^2\right )^{-1+3 \sqrt [4]{e}} \, dx+\left (4740 \sqrt [4]{e}\right ) \int (1-2 x) \left (x-x^2\right )^{-1+2 \sqrt [4]{e}} \, dx+\left (6160 \sqrt [4]{e}\right ) \int (1-2 x) \left (x-x^2\right )^{-1+\sqrt [4]{e}} \, dx\\ &=x+6160 \left (x-x^2\right )^{\sqrt [4]{e}}+2370 \left (x-x^2\right )^{2 \sqrt [4]{e}}+400 \left (x-x^2\right )^{3 \sqrt [4]{e}}+25 \left (x-x^2\right )^{4 \sqrt [4]{e}}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.13, size = 64, normalized size = 2.56 \begin {gather*} x+6160 (-((-1+x) x))^{\sqrt [4]{e}}+2370 (-((-1+x) x))^{2 \sqrt [4]{e}}+400 (-((-1+x) x))^{3 \sqrt [4]{e}}+25 (-((-1+x) x))^{4 \sqrt [4]{e}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 56, normalized size = 2.24 \begin {gather*} 25 \, {\left (-x^{2} + x\right )}^{4 \, e^{\frac {1}{4}}} + 400 \, {\left (-x^{2} + x\right )}^{3 \, e^{\frac {1}{4}}} + 2370 \, {\left (-x^{2} + x\right )}^{2 \, e^{\frac {1}{4}}} + 6160 \, {\left (-x^{2} + x\right )}^{e^{\frac {1}{4}}} + x \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 {100 \, {\left (-x^{2} + x\right )}^{4 \, e^{\frac {1}{4}}} {\left (2 \, x - 1\right )} e^{\frac {1}{4}} + 1200 \, {\left (-x^{2} + x\right )}^{3 \, e^{\frac {1}{4}}} {\left (2 \, x - 1\right )} e^{\frac {1}{4}} + 4740 \, {\left (-x^{2} + x\right )}^{2 \, e^{\frac {1}{4}}} {\left (2 \, x - 1\right )} e^{\frac {1}{4}} + 6160 \, {\left (-x^{2} + x\right )}^{e^{\frac {1}{4}}} {\left (2 \, x - 1\right )} e^{\frac {1}{4}} + x^{2} - x}{x^{2} - x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 57, normalized size = 2.28
method | result | size |
risch | \(25 \left (-x^{2}+x \right )^{4 \,{\mathrm e}^{\frac {1}{4}}}+400 \left (-x^{2}+x \right )^{3 \,{\mathrm e}^{\frac {1}{4}}}+2370 \left (-x^{2}+x \right )^{2 \,{\mathrm e}^{\frac {1}{4}}}+x +6160 \left (-x^{2}+x \right )^{{\mathrm e}^{\frac {1}{4}}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 80, normalized size = 3.20 \begin {gather*} x + 25 \, e^{\left (4 \, e^{\frac {1}{4}} \log \relax (x) + 4 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 400 \, e^{\left (3 \, e^{\frac {1}{4}} \log \relax (x) + 3 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 2370 \, e^{\left (2 \, e^{\frac {1}{4}} \log \relax (x) + 2 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 6160 \, e^{\left (e^{\frac {1}{4}} \log \relax (x) + e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.08, size = 56, normalized size = 2.24 \begin {gather*} x+2370\,{\left (x-x^2\right )}^{2\,{\mathrm {e}}^{1/4}}+400\,{\left (x-x^2\right )}^{3\,{\mathrm {e}}^{1/4}}+25\,{\left (x-x^2\right )}^{4\,{\mathrm {e}}^{1/4}}+6160\,{\left (x-x^2\right )}^{{\mathrm {e}}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.61, size = 53, normalized size = 2.12 \begin {gather*} x + 25 \left (- x^{2} + x\right )^{4 e^{\frac {1}{4}}} + 400 \left (- x^{2} + x\right )^{3 e^{\frac {1}{4}}} + 2370 \left (- x^{2} + x\right )^{2 e^{\frac {1}{4}}} + 6160 \left (- x^{2} + x\right )^{e^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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