Optimal. Leaf size=21 \[ 5+e^{x+\frac {1}{5} x \log (3)} (x-16 (1+x)) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(21)=42\).
time = 0.05, antiderivative size = 52, normalized size of antiderivative = 2.48, number of steps
used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {12, 2218, 2207,
2225} \begin {gather*} \frac {25\ 3^{\frac {x}{5}+1} e^x}{5+\log (3)}-\frac {3^{x/5} e^x (15 x (5+\log (3))+155+16 \log (3))}{5+\log (3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2218
Rule 2225
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int e^{\frac {1}{5} (5 x+x \log (3))} (-155-75 x+(-16-15 x) \log (3)) \, dx\\ &=\frac {1}{5} \int e^{\frac {1}{5} x (5+\log (3))} (-155-16 \log (3)-15 x (5+\log (3))) \, dx\\ &=-\frac {3^{x/5} e^x (155+16 \log (3)+15 x (5+\log (3)))}{5+\log (3)}+15 \int e^{\frac {1}{5} x (5+\log (3))} \, dx\\ &=\frac {25\ 3^{1+\frac {x}{5}} e^x}{5+\log (3)}-\frac {3^{x/5} e^x (155+16 \log (3)+15 x (5+\log (3)))}{5+\log (3)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 19, normalized size = 0.90 \begin {gather*} -\frac {1}{5} 3^{x/5} e^x (80+75 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(118\) vs.
\(2(16)=32\).
time = 0.04, size = 119, normalized size = 5.67
method | result | size |
risch | \(\frac {\left (-75 x -80\right ) 3^{\frac {x}{5}} {\mathrm e}^{x}}{5}\) | \(15\) |
gosper | \(-{\mathrm e}^{\frac {x \ln \left (3\right )}{5}+x} \left (15 x +16\right )\) | \(16\) |
norman | \(-15 x \,{\mathrm e}^{\frac {x \ln \left (3\right )}{5}+x}-16 \,{\mathrm e}^{\frac {x \ln \left (3\right )}{5}+x}\) | \(23\) |
meijerg | \(-\frac {155 \left (1-{\mathrm e}^{-\frac {x \left (-\ln \left (3\right )-5\right )}{5}}\right )}{-\ln \left (3\right )-5}+\frac {25 \left (-3 \ln \left (3\right )-15\right ) \left (1-\frac {\left (\frac {2 x \left (-\ln \left (3\right )-5\right )}{5}+2\right ) {\mathrm e}^{-\frac {x \left (-\ln \left (3\right )-5\right )}{5}}}{2}\right )}{\left (-\ln \left (3\right )-5\right )^{2}}-\frac {16 \ln \left (3\right ) \left (1-{\mathrm e}^{-\frac {x \left (-\ln \left (3\right )-5\right )}{5}}\right )}{-\ln \left (3\right )-5}\) | \(93\) |
derivativedivides | \(\frac {-16 \,{\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x} \ln \left (3\right )-\frac {375 \,{\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x} \left (\frac {\ln \left (3\right )}{5}+1\right ) x}{5+\ln \left (3\right )}+\frac {375 \,{\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x}}{5+\ln \left (3\right )}-\frac {75 \,{\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x} \ln \left (3\right ) \left (\frac {\ln \left (3\right )}{5}+1\right ) x}{5+\ln \left (3\right )}+\frac {75 \ln \left (3\right ) {\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x}}{5+\ln \left (3\right )}-155 \,{\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x}}{5+\ln \left (3\right )}\) | \(119\) |
default | \(\frac {-16 \,{\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x} \ln \left (3\right )-\frac {375 \,{\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x} \left (\frac {\ln \left (3\right )}{5}+1\right ) x}{5+\ln \left (3\right )}+\frac {375 \,{\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x}}{5+\ln \left (3\right )}-\frac {75 \,{\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x} \ln \left (3\right ) \left (\frac {\ln \left (3\right )}{5}+1\right ) x}{5+\ln \left (3\right )}+\frac {75 \ln \left (3\right ) {\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x}}{5+\ln \left (3\right )}-155 \,{\mathrm e}^{\left (\frac {\ln \left (3\right )}{5}+1\right ) x}}{5+\ln \left (3\right )}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs.
\(2 (17) = 34\).
time = 0.46, size = 97, normalized size = 4.62 \begin {gather*} -\frac {15 \, {\left (x {\left (\log \left (3\right ) + 5\right )} - 5\right )} e^{\left (\frac {1}{5} \, x \log \left (3\right ) + x\right )} \log \left (3\right )}{\log \left (3\right )^{2} + 10 \, \log \left (3\right ) + 25} - \frac {75 \, {\left (x {\left (\log \left (3\right ) + 5\right )} - 5\right )} e^{\left (\frac {1}{5} \, x \log \left (3\right ) + x\right )}}{\log \left (3\right )^{2} + 10 \, \log \left (3\right ) + 25} - \frac {16 \, e^{\left (\frac {1}{5} \, x \log \left (3\right ) + x\right )} \log \left (3\right )}{\log \left (3\right ) + 5} - \frac {155 \, e^{\left (\frac {1}{5} \, x \log \left (3\right ) + x\right )}}{\log \left (3\right ) + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 15, normalized size = 0.71 \begin {gather*} -{\left (15 \, x + 16\right )} e^{\left (\frac {1}{5} \, x \log \left (3\right ) + x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 15, normalized size = 0.71 \begin {gather*} \left (- 15 x - 16\right ) e^{\frac {x \log {\left (3 \right )}}{5} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (17) = 34\).
time = 0.40, size = 49, normalized size = 2.33 \begin {gather*} -\frac {{\left (15 \, x \log \left (3\right )^{2} + 150 \, x \log \left (3\right ) + 16 \, \log \left (3\right )^{2} + 375 \, x + 160 \, \log \left (3\right ) + 400\right )} e^{\left (\frac {1}{5} \, x \log \left (3\right ) + x\right )}}{\log \left (3\right )^{2} + 10 \, \log \left (3\right ) + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 14, normalized size = 0.67 \begin {gather*} -3^{x/5}\,{\mathrm {e}}^x\,\left (15\,x+16\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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