Optimal. Leaf size=22 \[ \left (\frac {6+2 x}{x (1+x)}\right )^{-x+x^2} \]
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Rubi [F] time = 2.93, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (\frac {6+2 x}{x+x^2}\right )^{-x+x^2} \left (3+3 x-5 x^2-x^3+\left (-3+2 x+7 x^2+2 x^3\right ) \log \left (\frac {6+2 x}{x+x^2}\right )\right )}{3+4 x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{3+4 x+x^2}+\frac {3 x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{3+4 x+x^2}-\frac {5 x^2 \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{3+4 x+x^2}-\frac {x^3 \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{3+4 x+x^2}+(-1+2 x) \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \log \left (\frac {6+2 x}{x (1+x)}\right )\right ) \, dx\\ &=3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{3+4 x+x^2} \, dx+3 \int \frac {x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{3+4 x+x^2} \, dx-5 \int \frac {x^2 \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{3+4 x+x^2} \, dx-\int \frac {x^3 \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{3+4 x+x^2} \, dx+\int (-1+2 x) \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \log \left (\frac {6+2 x}{x (1+x)}\right ) \, dx\\ &=3 \int \left (-\frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{-2-2 x}-\frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x}\right ) \, dx+3 \int \left (-\frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x}+\frac {3 \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x}\right ) \, dx-5 \int \left (\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}-\frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} (3+4 x)}{3+4 x+x^2}\right ) \, dx-\log \left (\frac {6+2 x}{x (1+x)}\right ) \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx+\left (2 \log \left (\frac {6+2 x}{x (1+x)}\right )\right ) \int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-\int \left (-4 \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}+x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}+\frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} (12+13 x)}{3+4 x+x^2}\right ) \, dx-\int \frac {\left (3+6 x+x^2\right ) \left (\int \left (\frac {6+2 x}{x+x^2}\right )^{(-1+x) x} \, dx-2 \int x \left (\frac {6+2 x}{x+x^2}\right )^{(-1+x) x} \, dx\right )}{x (1+x) (3+x)} \, dx\\ &=-\left (3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{-2-2 x} \, dx\right )-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x} \, dx-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx+4 \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-5 \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx+5 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} (3+4 x)}{3+4 x+x^2} \, dx+9 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx-\log \left (\frac {6+2 x}{x (1+x)}\right ) \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx+\left (2 \log \left (\frac {6+2 x}{x (1+x)}\right )\right ) \int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-\int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-\int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} (12+13 x)}{3+4 x+x^2} \, dx-\int \left (\frac {\left (3+6 x+x^2\right ) \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{x (1+x) (3+x)}+\frac {2 \left (-3-6 x-x^2\right ) \int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{x (1+x) (3+x)}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-3-6 x-x^2\right ) \int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{x (1+x) (3+x)} \, dx\right )-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{-2-2 x} \, dx-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x} \, dx-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx+4 \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-5 \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx+5 \int \left (-\frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x}+\frac {9 \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x}\right ) \, dx+9 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx-\log \left (\frac {6+2 x}{x (1+x)}\right ) \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx+\left (2 \log \left (\frac {6+2 x}{x (1+x)}\right )\right ) \int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-\int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-\int \left (-\frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x}+\frac {27 \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x}\right ) \, dx-\int \frac {\left (3+6 x+x^2\right ) \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{x (1+x) (3+x)} \, dx\\ &=-\left (2 \int \left (\frac {\int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{-1-x}-\frac {\int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{x}+\frac {\int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{3+x}\right ) \, dx\right )-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{-2-2 x} \, dx-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x} \, dx-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx+4 \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-5 \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-5 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x} \, dx+9 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx-27 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx+45 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx-\log \left (\frac {6+2 x}{x (1+x)}\right ) \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx+\left (2 \log \left (\frac {6+2 x}{x (1+x)}\right )\right ) \int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-\int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx+\int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x} \, dx-\int \left (\frac {\int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{-3-x}+\frac {\int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{x}+\frac {\int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{1+x}\right ) \, dx\\ &=-\left (2 \int \frac {\int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{-1-x} \, dx\right )+2 \int \frac {\int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{x} \, dx-2 \int \frac {\int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{3+x} \, dx-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{-2-2 x} \, dx-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x} \, dx-3 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx+4 \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-5 \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-5 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x} \, dx+9 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx-27 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx+45 \int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{6+2 x} \, dx-\log \left (\frac {6+2 x}{x (1+x)}\right ) \int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx+\left (2 \log \left (\frac {6+2 x}{x (1+x)}\right )\right ) \int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx-\int x \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx+\int \frac {\left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x}}{2+2 x} \, dx-\int \frac {\int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{-3-x} \, dx-\int \frac {\int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{x} \, dx-\int \frac {\int \left (\frac {6+2 x}{x (1+x)}\right )^{(-1+x) x} \, dx}{1+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.25, size = 19, normalized size = 0.86 \begin {gather*} \left (\frac {6+2 x}{x+x^2}\right )^{(-1+x) x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 20, normalized size = 0.91 \begin {gather*} \left (\frac {2 \, {\left (x + 3\right )}}{x^{2} + x}\right )^{x^{2} - 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 {{\left (x^{3} + 5 \, x^{2} - {\left (2 \, x^{3} + 7 \, x^{2} + 2 \, x - 3\right )} \log \left (\frac {2 \, {\left (x + 3\right )}}{x^{2} + x}\right ) - 3 \, x - 3\right )} \left (\frac {2 \, {\left (x + 3\right )}}{x^{2} + x}\right )^{x^{2} - x}}{x^{2} + 4 \, x + 3}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 20, normalized size = 0.91
method | result | size |
risch | \(\left (\frac {2 x +6}{x^{2}+x}\right )^{x \left (x -1\right )}\) | \(20\) |
norman | \({\mathrm e}^{\left (x^{2}-x \right ) \ln \left (\frac {2 x +6}{x^{2}+x}\right )}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 54, normalized size = 2.45 \begin {gather*} e^{\left (x^{2} \log \relax (2) + x^{2} \log \left (x + 3\right ) - x^{2} \log \left (x + 1\right ) - x^{2} \log \relax (x) - x \log \relax (2) - x \log \left (x + 3\right ) + x \log \left (x + 1\right ) + x \log \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.75, size = 21, normalized size = 0.95 \begin {gather*} {\left (\frac {2\,x+6}{x^2+x}\right )}^{x^2-x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 17, normalized size = 0.77 \begin {gather*} e^{\left (x^{2} - x\right ) \log {\left (\frac {2 x + 6}{x^{2} + x} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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