Optimal. Leaf size=30 \[ \frac {e^x+e^{5 x}-(5-x) \left (-2+e^x+x-\log (4)\right )}{x} \]
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Rubi [A]
time = 0.11, antiderivative size = 31, normalized size of antiderivative = 1.03, number of steps
used = 11, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {14, 2230,
2225, 2208, 2209, 2228} \begin {gather*} x+e^x-\frac {4 e^x}{x}+\frac {e^{5 x}}{x}+\frac {5 (2+\log (4))}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2208
Rule 2209
Rule 2225
Rule 2228
Rule 2230
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^x (-2+x)^2}{x^2}+\frac {e^{5 x} (-1+5 x)}{x^2}+\frac {-10+x^2-5 \log (4)}{x^2}\right ) \, dx\\ &=\int \frac {e^x (-2+x)^2}{x^2} \, dx+\int \frac {e^{5 x} (-1+5 x)}{x^2} \, dx+\int \frac {-10+x^2-5 \log (4)}{x^2} \, dx\\ &=\frac {e^{5 x}}{x}+\int \left (e^x+\frac {4 e^x}{x^2}-\frac {4 e^x}{x}\right ) \, dx+\int \left (1-\frac {5 (2+\log (4))}{x^2}\right ) \, dx\\ &=\frac {e^{5 x}}{x}+x+\frac {5 (2+\log (4))}{x}+4 \int \frac {e^x}{x^2} \, dx-4 \int \frac {e^x}{x} \, dx+\int e^x \, dx\\ &=e^x-\frac {4 e^x}{x}+\frac {e^{5 x}}{x}+x-4 \text {Ei}(x)+\frac {5 (2+\log (4))}{x}+4 \int \frac {e^x}{x} \, dx\\ &=e^x-\frac {4 e^x}{x}+\frac {e^{5 x}}{x}+x+\frac {5 (2+\log (4))}{x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 26, normalized size = 0.87 \begin {gather*} \frac {e^{5 x}+e^x (-4+x)+x^2+5 (2+\log (4))}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 32, normalized size = 1.07
method | result | size |
norman | \(\frac {x^{2}+{\mathrm e}^{5 x}+{\mathrm e}^{x} x -4 \,{\mathrm e}^{x}+10 \ln \left (2\right )+10}{x}\) | \(26\) |
default | \(x +\frac {10}{x}+\frac {{\mathrm e}^{5 x}}{x}-\frac {4 \,{\mathrm e}^{x}}{x}+\frac {10 \ln \left (2\right )}{x}+{\mathrm e}^{x}\) | \(32\) |
risch | \(x +\frac {10 \ln \left (2\right )}{x}+\frac {10}{x}+\frac {{\mathrm e}^{5 x}}{x}+\frac {\left (x -4\right ) {\mathrm e}^{x}}{x}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.31, size = 40, normalized size = 1.33 \begin {gather*} x + \frac {10 \, \log \left (2\right )}{x} + \frac {10}{x} + 5 \, {\rm Ei}\left (5 \, x\right ) - 4 \, {\rm Ei}\left (x\right ) + e^{x} + 4 \, \Gamma \left (-1, -x\right ) - 5 \, \Gamma \left (-1, -5 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 23, normalized size = 0.77 \begin {gather*} \frac {x^{2} + {\left (x - 4\right )} e^{x} + e^{\left (5 \, x\right )} + 10 \, \log \left (2\right ) + 10}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 29, normalized size = 0.97 \begin {gather*} x + \frac {10 \log {\left (2 \right )} + 10}{x} + \frac {x e^{5 x} + \left (x^{2} - 4 x\right ) e^{x}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 25, normalized size = 0.83 \begin {gather*} \frac {x^{2} + x e^{x} + e^{\left (5 \, x\right )} - 4 \, e^{x} + 10 \, \log \left (2\right ) + 10}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 22, normalized size = 0.73 \begin {gather*} x+{\mathrm {e}}^x+\frac {{\mathrm {e}}^{5\,x}+10\,\ln \left (2\right )-4\,{\mathrm {e}}^x+10}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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