Optimal. Leaf size=29 \[ e^{-2 x+\frac {e^x (5+x) \log (5)}{3 (-5+2 x)}}-x \]
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Rubi [B] time = 2.68, antiderivative size = 99, normalized size of antiderivative = 3.41, number of steps used = 5, number of rules used = 4, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {27, 12, 6742, 2288} \begin {gather*} \frac {5^{-\frac {e^x (x+5)}{3 (5-2 x)}} e^{-2 x} \left (-2 e^x x^2-5 e^x x+40 e^x\right )}{(5-2 x)^2 \left (\frac {e^x (x+5)}{5-2 x}+\frac {2 e^x (x+5)}{(5-2 x)^2}+\frac {e^x}{5-2 x}\right )}-x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 2288
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-75+60 x-12 x^2+\exp \left (\frac {30 x-12 x^2+e^x (5+x) \log (5)}{-15+6 x}\right ) \left (-150+120 x-24 x^2+e^x \left (-40+5 x+2 x^2\right ) \log (5)\right )}{3 (-5+2 x)^2} \, dx\\ &=\frac {1}{3} \int \frac {-75+60 x-12 x^2+\exp \left (\frac {30 x-12 x^2+e^x (5+x) \log (5)}{-15+6 x}\right ) \left (-150+120 x-24 x^2+e^x \left (-40+5 x+2 x^2\right ) \log (5)\right )}{(-5+2 x)^2} \, dx\\ &=\frac {1}{3} \int \left (-3+\frac {5^{\frac {e^x (5+x)}{-15+6 x}} e^{-2 x} \left (-150+120 x-24 x^2-40 e^x \log (5)+5 e^x x \log (5)+2 e^x x^2 \log (5)\right )}{(-5+2 x)^2}\right ) \, dx\\ &=-x+\frac {1}{3} \int \frac {5^{\frac {e^x (5+x)}{-15+6 x}} e^{-2 x} \left (-150+120 x-24 x^2-40 e^x \log (5)+5 e^x x \log (5)+2 e^x x^2 \log (5)\right )}{(-5+2 x)^2} \, dx\\ &=-x+\frac {5^{-\frac {e^x (5+x)}{3 (5-2 x)}} e^{-2 x} \left (40 e^x-5 e^x x-2 e^x x^2\right )}{(5-2 x)^2 \left (\frac {e^x}{5-2 x}+\frac {2 e^x (5+x)}{(5-2 x)^2}+\frac {e^x (5+x)}{5-2 x}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.75, size = 31, normalized size = 1.07 \begin {gather*} \frac {1}{3} \left (3\ 5^{\frac {e^x (5+x)}{-15+6 x}} e^{-2 x}-3 x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 31, normalized size = 1.07 \begin {gather*} -x + e^{\left (\frac {{\left (x + 5\right )} e^{x} \log \relax (5) - 12 \, x^{2} + 30 \, x}{3 \, {\left (2 \, x - 5\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {12 \, x^{2} - {\left ({\left (2 \, x^{2} + 5 \, x - 40\right )} e^{x} \log \relax (5) - 24 \, x^{2} + 120 \, x - 150\right )} e^{\left (\frac {{\left (x + 5\right )} e^{x} \log \relax (5) - 12 \, x^{2} + 30 \, x}{3 \, {\left (2 \, x - 5\right )}}\right )} - 60 \, x + 75}{3 \, {\left (4 \, x^{2} - 20 \, x + 25\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 37, normalized size = 1.28
method | result | size |
risch | \(-x +5^{\frac {x \,{\mathrm e}^{x}}{6 x -15}} 5^{\frac {5 \,{\mathrm e}^{x}}{3 \left (2 x -5\right )}} {\mathrm e}^{-2 x}\) | \(37\) |
norman | \(\frac {-2 x^{2}+2 x \,{\mathrm e}^{\frac {\left (5+x \right ) \ln \relax (5) {\mathrm e}^{x}-12 x^{2}+30 x}{6 x -15}}-5 \,{\mathrm e}^{\frac {\left (5+x \right ) \ln \relax (5) {\mathrm e}^{x}-12 x^{2}+30 x}{6 x -15}}+\frac {25}{2}}{2 x -5}\) | \(73\) |
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*} -x + \frac {1}{3} \, \int -\frac {{\left (24 \, x^{2} - {\left (2 \, x^{2} \log \relax (5) + 5 \, x \log \relax (5) - 40 \, \log \relax (5)\right )} e^{x} - 120 \, x + 150\right )} e^{\left (\frac {1}{6} \, e^{x} \log \relax (5) - 2 \, x + \frac {5 \, e^{x} \log \relax (5)}{2 \, {\left (2 \, x - 5\right )}}\right )}}{4 \, x^{2} - 20 \, x + 25}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.57, size = 55, normalized size = 1.90 \begin {gather*} 5^{\frac {5\,{\mathrm {e}}^x}{6\,x-15}}\,5^{\frac {x\,{\mathrm {e}}^x}{6\,x-15}}\,{\mathrm {e}}^{-\frac {12\,x^2}{6\,x-15}}\,{\mathrm {e}}^{\frac {30\,x}{6\,x-15}}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 26, normalized size = 0.90 \begin {gather*} - x + e^{\frac {- 12 x^{2} + 30 x + \left (x + 5\right ) e^{x} \log {\relax (5 )}}{6 x - 15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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