Optimal. Leaf size=18 \[ 3 e^x \log \left (4+\frac {e^{5 x}}{4}+x\right ) \]
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Rubi [A] time = 1.00, antiderivative size = 20, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 3, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6742, 2194, 2554} \begin {gather*} 3 e^x \log \left (\frac {1}{4} \left (4 x+e^{5 x}+16\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2194
Rule 2554
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {12 e^x (19+5 x)}{16+e^{5 x}+4 x}+3 e^x \left (5+\log \left (4+\frac {e^{5 x}}{4}+x\right )\right )\right ) \, dx\\ &=3 \int e^x \left (5+\log \left (4+\frac {e^{5 x}}{4}+x\right )\right ) \, dx-12 \int \frac {e^x (19+5 x)}{16+e^{5 x}+4 x} \, dx\\ &=3 \int \left (5 e^x+e^x \log \left (\frac {1}{4} \left (16+e^{5 x}+4 x\right )\right )\right ) \, dx-12 \int \left (\frac {19 e^x}{16+e^{5 x}+4 x}+\frac {5 e^x x}{16+e^{5 x}+4 x}\right ) \, dx\\ &=3 \int e^x \log \left (\frac {1}{4} \left (16+e^{5 x}+4 x\right )\right ) \, dx+15 \int e^x \, dx-60 \int \frac {e^x x}{16+e^{5 x}+4 x} \, dx-228 \int \frac {e^x}{16+e^{5 x}+4 x} \, dx\\ &=15 e^x+3 e^x \log \left (\frac {1}{4} \left (16+e^{5 x}+4 x\right )\right )-3 \int \frac {e^x \left (4+5 e^{5 x}\right )}{e^{5 x}+4 (4+x)} \, dx-60 \int \frac {e^x x}{16+e^{5 x}+4 x} \, dx-228 \int \frac {e^x}{16+e^{5 x}+4 x} \, dx\\ &=15 e^x+3 e^x \log \left (\frac {1}{4} \left (16+e^{5 x}+4 x\right )\right )-3 \int \left (5 e^x-\frac {4 e^x (19+5 x)}{16+e^{5 x}+4 x}\right ) \, dx-60 \int \frac {e^x x}{16+e^{5 x}+4 x} \, dx-228 \int \frac {e^x}{16+e^{5 x}+4 x} \, dx\\ &=15 e^x+3 e^x \log \left (\frac {1}{4} \left (16+e^{5 x}+4 x\right )\right )+12 \int \frac {e^x (19+5 x)}{16+e^{5 x}+4 x} \, dx-15 \int e^x \, dx-60 \int \frac {e^x x}{16+e^{5 x}+4 x} \, dx-228 \int \frac {e^x}{16+e^{5 x}+4 x} \, dx\\ &=3 e^x \log \left (\frac {1}{4} \left (16+e^{5 x}+4 x\right )\right )+12 \int \left (\frac {19 e^x}{16+e^{5 x}+4 x}+\frac {5 e^x x}{16+e^{5 x}+4 x}\right ) \, dx-60 \int \frac {e^x x}{16+e^{5 x}+4 x} \, dx-228 \int \frac {e^x}{16+e^{5 x}+4 x} \, dx\\ &=3 e^x \log \left (\frac {1}{4} \left (16+e^{5 x}+4 x\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 18, normalized size = 1.00 \begin {gather*} 3 e^x \log \left (4+\frac {e^{5 x}}{4}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 14, normalized size = 0.78 \begin {gather*} 3 \, e^{x} \log \left (x + \frac {1}{4} \, e^{\left (5 \, x\right )} + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 21, normalized size = 1.17 \begin {gather*} -6 \, e^{x} \log \relax (2) + 3 \, e^{x} \log \left (4 \, x + e^{\left (5 \, x\right )} + 16\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 15, normalized size = 0.83
method | result | size |
risch | \(3 \ln \left (\frac {{\mathrm e}^{5 x}}{4}+x +4\right ) {\mathrm e}^{x}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 21, normalized size = 1.17 \begin {gather*} -6 \, e^{x} \log \relax (2) + 3 \, e^{x} \log \left (4 \, x + e^{\left (5 \, x\right )} + 16\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 14, normalized size = 0.78 \begin {gather*} 3\,{\mathrm {e}}^x\,\ln \left (x+\frac {{\mathrm {e}}^{5\,x}}{4}+4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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