Optimal. Leaf size=27 \[ \frac {16 \left (3+\frac {5-e^x}{x}\right ) x^2}{5 (5-x)} \]
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Rubi [A]
time = 0.12, antiderivative size = 34, normalized size of antiderivative = 1.26, number of steps
used = 12, number of rules used = 8, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {27, 12, 6874,
2230, 2225, 2208, 2209, 697} \begin {gather*} -\frac {48 x}{5}+\frac {16 e^x}{5}-\frac {16 e^x}{5-x}+\frac {320}{5-x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 697
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {400+480 x-48 x^2+e^x \left (-80-80 x+16 x^2\right )}{5 (-5+x)^2} \, dx\\ &=\frac {1}{5} \int \frac {400+480 x-48 x^2+e^x \left (-80-80 x+16 x^2\right )}{(-5+x)^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {16 e^x \left (-5-5 x+x^2\right )}{(-5+x)^2}-\frac {16 \left (-25-30 x+3 x^2\right )}{(-5+x)^2}\right ) \, dx\\ &=\frac {16}{5} \int \frac {e^x \left (-5-5 x+x^2\right )}{(-5+x)^2} \, dx-\frac {16}{5} \int \frac {-25-30 x+3 x^2}{(-5+x)^2} \, dx\\ &=-\left (\frac {16}{5} \int \left (3-\frac {100}{(-5+x)^2}\right ) \, dx\right )+\frac {16}{5} \int \left (e^x-\frac {5 e^x}{(-5+x)^2}+\frac {5 e^x}{-5+x}\right ) \, dx\\ &=\frac {320}{5-x}-\frac {48 x}{5}+\frac {16 \int e^x \, dx}{5}-16 \int \frac {e^x}{(-5+x)^2} \, dx+16 \int \frac {e^x}{-5+x} \, dx\\ &=\frac {16 e^x}{5}+\frac {320}{5-x}-\frac {16 e^x}{5-x}-\frac {48 x}{5}+16 e^5 \text {Ei}(-5+x)-16 \int \frac {e^x}{-5+x} \, dx\\ &=\frac {16 e^x}{5}+\frac {320}{5-x}-\frac {16 e^x}{5-x}-\frac {48 x}{5}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.12, size = 23, normalized size = 0.85 \begin {gather*} \frac {16 \left (-100+\left (15+e^x\right ) x-3 x^2\right )}{5 (-5+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 25, normalized size = 0.93
method | result | size |
norman | \(\frac {-\frac {48 x^{2}}{5}+\frac {16 \,{\mathrm e}^{x} x}{5}-80}{x -5}\) | \(19\) |
risch | \(-\frac {48 x}{5}-\frac {320}{x -5}+\frac {16 x \,{\mathrm e}^{x}}{5 \left (x -5\right )}\) | \(22\) |
default | \(-\frac {320}{x -5}-\frac {48 x}{5}+\frac {16 \,{\mathrm e}^{x}}{x -5}+\frac {16 \,{\mathrm e}^{x}}{5}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 = 0.81 \begin {gather*} -\frac {16 \, {\left (3 \, x^{2} - x e^{x} - 15 \, x + 100\right )}}{5 \, {\left (x - 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 20, normalized size = 0.74 \begin {gather*} - \frac {48 x}{5} + \frac {16 x e^{x}}{5 x - 25} - \frac {320}{x - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.37, size = 22, normalized size = 0.81 \begin {gather*} -\frac {16 \, {\left (3 \, x^{2} - x e^{x} - 15 \, x + 100\right )}}{5 \, {\left (x - 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 19, normalized size = 0.70 \begin {gather*} -\frac {16\,x\,\left (3\,x-{\mathrm {e}}^x+5\right )}{5\,\left (x-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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