Optimal. Leaf size=22 \[ \frac {x^2}{e^x+625 x+e^{-x} x+\log (x)} \]
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Rubi [F] time = 6.95, 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 {e^{3 x} \left (2 x-x^2\right )+e^{2 x} \left (-x+625 x^2\right )+e^x \left (x^2+x^3\right )+2 e^{2 x} x \log (x)}{e^{4 x}+1250 e^{3 x} x+x^2+1250 e^x x^2+e^{2 x} \left (2 x+390625 x^2\right )+\left (2 e^{3 x}+2 e^x x+1250 e^{2 x} x\right ) \log (x)+e^{2 x} \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x x \left (-e^x (1-625 x)-e^{2 x} (-2+x)+x (1+x)+2 e^x \log (x)\right )}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx\\ &=\int \left (-\frac {e^x (-2+x) x}{e^{2 x}+x+625 e^x x+e^x \log (x)}+\frac {e^x x \left (-e^x-x-625 e^x x+2 x^2+625 e^x x^2+e^x x \log (x)\right )}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}\right ) \, dx\\ &=-\int \frac {e^x (-2+x) x}{e^{2 x}+x+625 e^x x+e^x \log (x)} \, dx+\int \frac {e^x x \left (-e^x-x-625 e^x x+2 x^2+625 e^x x^2+e^x x \log (x)\right )}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx\\ &=\int \left (-\frac {e^{2 x} x}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}-\frac {e^x x^2}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}-\frac {625 e^{2 x} x^2}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}+\frac {2 e^x x^3}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}+\frac {625 e^{2 x} x^3}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}+\frac {e^{2 x} x^2 \log (x)}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2}\right ) \, dx-\int \left (-\frac {2 e^x x}{e^{2 x}+x+625 e^x x+e^x \log (x)}+\frac {e^x x^2}{e^{2 x}+x+625 e^x x+e^x \log (x)}\right ) \, dx\\ &=2 \int \frac {e^x x^3}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx+2 \int \frac {e^x x}{e^{2 x}+x+625 e^x x+e^x \log (x)} \, dx-625 \int \frac {e^{2 x} x^2}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx+625 \int \frac {e^{2 x} x^3}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx-\int \frac {e^{2 x} x}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx-\int \frac {e^x x^2}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx+\int \frac {e^{2 x} x^2 \log (x)}{\left (e^{2 x}+x+625 e^x x+e^x \log (x)\right )^2} \, dx-\int \frac {e^x x^2}{e^{2 x}+x+625 e^x x+e^x \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 2.95, size = 28, normalized size = 1.27 \begin {gather*} \frac {e^x x^2}{e^{2 x}+x+625 e^x x+e^x \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 24, normalized size = 1.09 \begin {gather*} \frac {x^{2} e^{x}}{625 \, x e^{x} + e^{x} \log \relax (x) + x + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 24, normalized size = 1.09 \begin {gather*} \frac {x^{2} e^{x}}{625 \, x e^{x} + e^{x} \log \relax (x) + x + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 1.14
method | result | size |
risch | \(\frac {{\mathrm e}^{x} x^{2}}{{\mathrm e}^{x} \ln \relax (x )+{\mathrm e}^{2 x}+625 \,{\mathrm e}^{x} x +x}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 23, normalized size = 1.05 \begin {gather*} \frac {x^{2} e^{x}}{{\left (625 \, x + \log \relax (x)\right )} e^{x} + x + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 98, normalized size = 4.45 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,\left (x^4-x^5\right )+x^3\,{\mathrm {e}}^{3\,x}+625\,x^4\,{\mathrm {e}}^{3\,x}+x^4\,{\mathrm {e}}^{4\,x}}{\left (x+{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\ln \relax (x)+625\,x\,{\mathrm {e}}^x\right )\,\left (x\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^x-x^3\,{\mathrm {e}}^x+625\,x^2\,{\mathrm {e}}^{2\,x}+x^2\,{\mathrm {e}}^{3\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 22, normalized size = 1.00 \begin {gather*} \frac {x^{2} e^{x}}{x + \left (625 x + \log {\relax (x )}\right ) e^{x} + e^{2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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