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] Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(25)=50\).
time = 1.83, 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 = {1607, 6874,
6820, 643} \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 643
Rule 1607
Rule 6820
Rule 6874
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] Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(25)=50\).
time = 5.23, 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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs.
\(2(24)=48\).
time = 0.46, size = 86, normalized size = 3.44
method | result | size |
risch | \(25 \left (-x^{2}+x \right )^{4 \,{\mathrm e}^{\frac {1}{4}}}+400 \left (-x^{2}+x \right )^{3 \,{\mathrm e}^{\frac {1}{4}}}+2370 \left (-x^{2}+x \right )^{2 \,{\mathrm e}^{\frac {1}{4}}}+x +6160 \left (-x^{2}+x \right )^{{\mathrm e}^{\frac {1}{4}}}\) | \(57\) |
default | \(x +6160 \,{\mathrm e}^{-\frac {1}{4}} {\mathrm e}^{\frac {1}{4}+{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+2370 \,{\mathrm e}^{-\frac {1}{4}} {\mathrm e}^{\frac {1}{4}+2 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+400 \,{\mathrm e}^{-\frac {1}{4}} {\mathrm e}^{\frac {1}{4}+3 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}+25 \,{\mathrm e}^{-\frac {1}{4}} {\mathrm e}^{\frac {1}{4}+4 \,{\mathrm e}^{\frac {1}{4}} \ln \left (-x^{2}+x \right )}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs.
\(2 (22) = 44\).
time = 0.31, size = 80, normalized size = 3.20 \begin {gather*} x + 25 \, e^{\left (4 \, e^{\frac {1}{4}} \log \left (x\right ) + 4 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 400 \, e^{\left (3 \, e^{\frac {1}{4}} \log \left (x\right ) + 3 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 2370 \, e^{\left (2 \, e^{\frac {1}{4}} \log \left (x\right ) + 2 \, e^{\frac {1}{4}} \log \left (-x + 1\right )\right )} + 6160 \, e^{\left (e^{\frac {1}{4}} \log \left (x\right ) + 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (22) = 44\).
time = 0.40, 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (19) = 38\).
time = 3.95, 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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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