Optimal. Leaf size=25 \[ e^{\frac {3+x}{2}}+x \log \left (36-3 e^x+\frac {4}{x}\right ) \]
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Rubi [A]
time = 0.69, antiderivative size = 28, normalized size of antiderivative = 1.12, number of steps
used = 13, number of rules used = 5, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6873, 12,
6874, 2225, 2628} \begin {gather*} e^{\frac {x}{2}+\frac {3}{2}}+x \log \left (4 \left (\frac {1}{x}+9\right )-3 e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2225
Rule 2628
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8-e^{\frac {3+x}{2}} (-4-36 x)-e^x \left (3 e^{\frac {3+x}{2}} x+6 x^2\right )-\left (-8-72 x+6 e^x x\right ) \log \left (\frac {4+36 x-3 e^x x}{x}\right )}{2 \left (4+36 x-3 e^x x\right )} \, dx\\ &=\frac {1}{2} \int \frac {-8-e^{\frac {3+x}{2}} (-4-36 x)-e^x \left (3 e^{\frac {3+x}{2}} x+6 x^2\right )-\left (-8-72 x+6 e^x x\right ) \log \left (\frac {4+36 x-3 e^x x}{x}\right )}{4+36 x-3 e^x x} \, dx\\ &=\frac {1}{2} \int \left (e^{\frac {3}{2}+\frac {x}{2}}+\frac {8 \left (1+x+9 x^2\right )}{-4-36 x+3 e^x x}+2 \left (x+\log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )\right )\right ) \, dx\\ &=\frac {1}{2} \int e^{\frac {3}{2}+\frac {x}{2}} \, dx+4 \int \frac {1+x+9 x^2}{-4-36 x+3 e^x x} \, dx+\int \left (x+\log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )\right ) \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+\frac {x^2}{2}+4 \int \left (\frac {1}{-4-36 x+3 e^x x}+\frac {x}{-4-36 x+3 e^x x}+\frac {9 x^2}{-4-36 x+3 e^x x}\right ) \, dx+\int \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right ) \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+\frac {x^2}{2}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )+4 \int \frac {1}{-4-36 x+3 e^x x} \, dx+4 \int \frac {x}{-4-36 x+3 e^x x} \, dx+36 \int \frac {x^2}{-4-36 x+3 e^x x} \, dx-\int \frac {-4-3 e^x x^2}{4-3 \left (-12+e^x\right ) x} \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+\frac {x^2}{2}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )+4 \int \frac {1}{-4-36 x+3 e^x x} \, dx+4 \int \frac {x}{-4-36 x+3 e^x x} \, dx+36 \int \frac {x^2}{-4-36 x+3 e^x x} \, dx-\int \left (x+\frac {4 \left (1+x+9 x^2\right )}{-4-36 x+3 e^x x}\right ) \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )+4 \int \frac {1}{-4-36 x+3 e^x x} \, dx+4 \int \frac {x}{-4-36 x+3 e^x x} \, dx-4 \int \frac {1+x+9 x^2}{-4-36 x+3 e^x x} \, dx+36 \int \frac {x^2}{-4-36 x+3 e^x x} \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )+4 \int \frac {1}{-4-36 x+3 e^x x} \, dx+4 \int \frac {x}{-4-36 x+3 e^x x} \, dx-4 \int \left (\frac {1}{-4-36 x+3 e^x x}+\frac {x}{-4-36 x+3 e^x x}+\frac {9 x^2}{-4-36 x+3 e^x x}\right ) \, dx+36 \int \frac {x^2}{-4-36 x+3 e^x x} \, dx\\ &=e^{\frac {3}{2}+\frac {x}{2}}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.19, size = 28, normalized size = 1.12 \begin {gather*} e^{\frac {3}{2}+\frac {x}{2}}+x \log \left (-3 e^x+4 \left (9+\frac {1}{x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.24, size = 179, normalized size = 7.16
method | result | size |
risch | \(x \ln \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )-x \ln \left (x \right )-i \pi x \mathrm {csgn}\left (\frac {i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )}{x}\right )^{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )\right ) \mathrm {csgn}\left (\frac {i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )}{x}\right )^{2}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )\right ) \mathrm {csgn}\left (\frac {i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )}{2}+\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )}{x}\right )^{3}}{2}+\frac {i \pi x \mathrm {csgn}\left (\frac {i \left (-\frac {4}{3}+\left ({\mathrm e}^{x}-12\right ) x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )}{2}+i x \pi +x \ln \left (3\right )+{\mathrm e}^{\frac {3}{2}+\frac {x}{2}}\) | \(179\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 25, normalized size = 1.00 \begin {gather*} x \log \left (-3 \, x e^{x} + 36 \, x + 4\right ) - x \log \left (x\right ) + e^{\left (\frac {1}{2} \, x + \frac {3}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 33, normalized size = 1.32 \begin {gather*} x \log \left (\frac {{\left (4 \, {\left (9 \, x + 1\right )} e^{3} - 3 \, x e^{\left (x + 3\right )}\right )} e^{\left (-3\right )}}{x}\right ) + e^{\left (\frac {1}{2} \, x + \frac {3}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 27, normalized size = 1.08 \begin {gather*} x \log {\left (\frac {- 3 x e^{x} + 36 x + 4}{x} \right )} + e^{\frac {3}{2}} \sqrt {e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 25, normalized size = 1.00 \begin {gather*} x \log \left (-\frac {3 \, x e^{x} - 36 \, x - 4}{x}\right ) + e^{\left (\frac {1}{2} \, x + \frac {3}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.37, size = 24, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{\frac {x}{2}+\frac {3}{2}}+x\,\ln \left (\frac {36\,x-3\,x\,{\mathrm {e}}^x+4}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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