Optimal. Leaf size=24 \[ x \left (3+\log \left (4 \left (-5+e^x x-\frac {1}{3} (2-x) x\right )\right )\right ) \]
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Rubi [A]
time = 1.20, antiderivative size = 27, normalized size of antiderivative = 1.12, number of steps
used = 9, number of rules used = 2, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6874, 2628}
\begin {gather*} x \log \left (-\frac {4}{3} \left (-x^2+\left (2-3 e^x\right ) x+15\right )\right )+3 x \end {gather*}
Antiderivative was successfully verified.
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Rule 2628
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4+x-\frac {-15-15 x-3 x^2+x^3}{-15-2 x+3 e^x x+x^2}+\log \left (\frac {4}{3} \left (-15+\left (-2+3 e^x\right ) x+x^2\right )\right )\right ) \, dx\\ &=4 x+\frac {x^2}{2}-\int \frac {-15-15 x-3 x^2+x^3}{-15-2 x+3 e^x x+x^2} \, dx+\int \log \left (\frac {4}{3} \left (-15+\left (-2+3 e^x\right ) x+x^2\right )\right ) \, dx\\ &=4 x+\frac {x^2}{2}+x \log \left (-\frac {4}{3} \left (15+\left (2-3 e^x\right ) x-x^2\right )\right )-\int \frac {x \left (-2 (-1+x)-3 e^x (1+x)\right )}{15-\left (-2+3 e^x\right ) x-x^2} \, dx-\int \left (-\frac {15}{-15-2 x+3 e^x x+x^2}-\frac {15 x}{-15-2 x+3 e^x x+x^2}-\frac {3 x^2}{-15-2 x+3 e^x x+x^2}+\frac {x^3}{-15-2 x+3 e^x x+x^2}\right ) \, dx\\ &=4 x+\frac {x^2}{2}+x \log \left (-\frac {4}{3} \left (15+\left (2-3 e^x\right ) x-x^2\right )\right )+3 \int \frac {x^2}{-15-2 x+3 e^x x+x^2} \, dx+15 \int \frac {1}{-15-2 x+3 e^x x+x^2} \, dx+15 \int \frac {x}{-15-2 x+3 e^x x+x^2} \, dx-\int \frac {x^3}{-15-2 x+3 e^x x+x^2} \, dx-\int \left (1+x-\frac {-15-15 x-3 x^2+x^3}{-15-2 x+3 e^x x+x^2}\right ) \, dx\\ &=3 x+x \log \left (-\frac {4}{3} \left (15+\left (2-3 e^x\right ) x-x^2\right )\right )+3 \int \frac {x^2}{-15-2 x+3 e^x x+x^2} \, dx+15 \int \frac {1}{-15-2 x+3 e^x x+x^2} \, dx+15 \int \frac {x}{-15-2 x+3 e^x x+x^2} \, dx-\int \frac {x^3}{-15-2 x+3 e^x x+x^2} \, dx+\int \frac {-15-15 x-3 x^2+x^3}{-15-2 x+3 e^x x+x^2} \, dx\\ &=3 x+x \log \left (-\frac {4}{3} \left (15+\left (2-3 e^x\right ) x-x^2\right )\right )+3 \int \frac {x^2}{-15-2 x+3 e^x x+x^2} \, dx+15 \int \frac {1}{-15-2 x+3 e^x x+x^2} \, dx+15 \int \frac {x}{-15-2 x+3 e^x x+x^2} \, dx-\int \frac {x^3}{-15-2 x+3 e^x x+x^2} \, dx+\int \left (-\frac {15}{-15-2 x+3 e^x x+x^2}-\frac {15 x}{-15-2 x+3 e^x x+x^2}-\frac {3 x^2}{-15-2 x+3 e^x x+x^2}+\frac {x^3}{-15-2 x+3 e^x x+x^2}\right ) \, dx\\ &=3 x+x \log \left (-\frac {4}{3} \left (15+\left (2-3 e^x\right ) x-x^2\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.14, size = 25, normalized size = 1.04 \begin {gather*} 3 x+x \log \left (\frac {4}{3} \left (-15+\left (-2+3 e^x\right ) x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 23, normalized size = 0.96
method | result | size |
norman | \(\ln \left (4 \,{\mathrm e}^{x} x +\frac {4 x^{2}}{3}-\frac {8 x}{3}-20\right ) x +3 x\) | \(23\) |
risch | \(\ln \left (4 \,{\mathrm e}^{x} x +\frac {4 x^{2}}{3}-\frac {8 x}{3}-20\right ) x +3 x\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 28, normalized size = 1.17 \begin {gather*} -x {\left (\log \left (3\right ) - 2 \, \log \left (2\right ) - 3\right )} + x \log \left (x^{2} + 3 \, x e^{x} - 2 \, x - 15\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 22, normalized size = 0.92 \begin {gather*} x \log \left (\frac {4}{3} \, x^{2} + 4 \, x e^{x} - \frac {8}{3} \, x - 20\right ) + 3 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 26, normalized size = 1.08 \begin {gather*} x \log {\left (\frac {4 x^{2}}{3} + 4 x e^{x} - \frac {8 x}{3} - 20 \right )} + 3 x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 22, normalized size = 0.92 \begin {gather*} x \log \left (\frac {4}{3} \, x^{2} + 4 \, x e^{x} - \frac {8}{3} \, x - 20\right ) + 3 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.56, size = 20, normalized size = 0.83 \begin {gather*} x\,\left (\ln \left (4\,x\,{\mathrm {e}}^x-\frac {8\,x}{3}+\frac {4\,x^2}{3}-20\right )+3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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