Optimal. Leaf size=17 \[ x^2+e^x \left (e^x+5 x\right ) \log (x) \]
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Rubi [B] time = 0.34, antiderivative size = 58, normalized size of antiderivative = 3.41, number of steps used = 14, number of rules used = 9, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.134, Rules used = {6688, 2194, 2178, 6742, 2176, 2554, 2262, 2177, 14} \begin {gather*} x^2+\frac {25 x}{2}+5 e^x+\frac {e^{2 x}}{2 x}-\frac {\left (5 x+e^x\right )^2}{2 x}+5 e^x x \log (x)+e^{2 x} \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2262
Rule 2554
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (5 e^x+\frac {e^{2 x}}{x}+2 x+e^x \left (5+2 e^x+5 x\right ) \log (x)\right ) \, dx\\ &=x^2+5 \int e^x \, dx+\int \frac {e^{2 x}}{x} \, dx+\int e^x \left (5+2 e^x+5 x\right ) \log (x) \, dx\\ &=5 e^x+x^2+\text {Ei}(2 x)+e^{2 x} \log (x)+5 e^x x \log (x)-\int \frac {e^x \left (e^x+5 x\right )}{x} \, dx\\ &=5 e^x+x^2-\frac {\left (e^x+5 x\right )^2}{2 x}+\text {Ei}(2 x)+e^{2 x} \log (x)+5 e^x x \log (x)-\frac {1}{2} \int \frac {\left (e^x+5 x\right )^2}{x^2} \, dx+5 \int \frac {e^x+5 x}{x} \, dx\\ &=5 e^x+x^2-\frac {\left (e^x+5 x\right )^2}{2 x}+\text {Ei}(2 x)+e^{2 x} \log (x)+5 e^x x \log (x)-\frac {1}{2} \int \left (25+\frac {e^{2 x}}{x^2}+\frac {10 e^x}{x}\right ) \, dx+5 \int \left (5+\frac {e^x}{x}\right ) \, dx\\ &=5 e^x+\frac {25 x}{2}+x^2-\frac {\left (e^x+5 x\right )^2}{2 x}+\text {Ei}(2 x)+e^{2 x} \log (x)+5 e^x x \log (x)-\frac {1}{2} \int \frac {e^{2 x}}{x^2} \, dx\\ &=5 e^x+\frac {e^{2 x}}{2 x}+\frac {25 x}{2}+x^2-\frac {\left (e^x+5 x\right )^2}{2 x}+\text {Ei}(2 x)+e^{2 x} \log (x)+5 e^x x \log (x)-\int \frac {e^{2 x}}{x} \, dx\\ &=5 e^x+\frac {e^{2 x}}{2 x}+\frac {25 x}{2}+x^2-\frac {\left (e^x+5 x\right )^2}{2 x}+e^{2 x} \log (x)+5 e^x x \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 17, normalized size = 1.00 \begin {gather*} x^2+e^x \left (e^x+5 x\right ) \log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.66, size = 17, normalized size = 1.00 \begin {gather*} x^{2} + {\left (5 \, x e^{x} + e^{\left (2 \, x\right )}\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 18, normalized size = 1.06 \begin {gather*} 5 \, x e^{x} \log \relax (x) + x^{2} + e^{\left (2 \, x\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 18, normalized size = 1.06
method | result | size |
risch | \(\left (5 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}\right ) \ln \relax (x )+x^{2}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 18, normalized size = 1.06 \begin {gather*} 5 \, x e^{x} \log \relax (x) + x^{2} + e^{\left (2 \, x\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.36, size = 18, normalized size = 1.06 \begin {gather*} {\mathrm {e}}^{2\,x}\,\ln \relax (x)+x^2+5\,x\,{\mathrm {e}}^x\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 20, normalized size = 1.18 \begin {gather*} x^{2} + 5 x e^{x} \log {\relax (x )} + e^{2 x} \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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