Optimal. Leaf size=19 \[ e^{-3+2 x-16 (x-\log (2))} x^3 \]
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Rubi [A]
time = 0.14, antiderivative size = 12, normalized size of antiderivative = 0.63, number of steps
used = 11, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {12, 1607,
2227, 2207, 2225} \begin {gather*} 65536 e^{-14 x-3} x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=65536 \int e^{-3-14 x} \left (3 x^2-14 x^3\right ) \, dx\\ &=65536 \int e^{-3-14 x} (3-14 x) x^2 \, dx\\ &=65536 \int \left (3 e^{-3-14 x} x^2-14 e^{-3-14 x} x^3\right ) \, dx\\ &=196608 \int e^{-3-14 x} x^2 \, dx-917504 \int e^{-3-14 x} x^3 \, dx\\ &=-\frac {98304}{7} e^{-3-14 x} x^2+65536 e^{-3-14 x} x^3+\frac {196608}{7} \int e^{-3-14 x} x \, dx-196608 \int e^{-3-14 x} x^2 \, dx\\ &=-\frac {98304}{49} e^{-3-14 x} x+65536 e^{-3-14 x} x^3+\frac {98304}{49} \int e^{-3-14 x} \, dx-\frac {196608}{7} \int e^{-3-14 x} x \, dx\\ &=-\frac {49152}{343} e^{-3-14 x}+65536 e^{-3-14 x} x^3-\frac {98304}{49} \int e^{-3-14 x} \, dx\\ &=65536 e^{-3-14 x} x^3\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 12, normalized size = 0.63 \begin {gather*} 65536 e^{-3-14 x} x^3 \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. \(68\) vs.
\(2(18)=36\).
time = 0.20, size = 69, normalized size = 3.63
method | result | size |
risch | \(65536 x^{3} {\mathrm e}^{-14 x -3}\) | \(12\) |
gosper | \(65536 x^{3} {\mathrm e}^{-15 x -3} {\mathrm e}^{x}\) | \(19\) |
default | \(196608 \,{\mathrm e}^{-3} \left (-\frac {{\mathrm e}^{-14 x} x^{2}}{14}-\frac {x \,{\mathrm e}^{-14 x}}{98}-\frac {{\mathrm e}^{-14 x}}{1372}\right )-917504 \,{\mathrm e}^{-3} \left (-\frac {{\mathrm e}^{-14 x} x^{3}}{14}-\frac {3 \,{\mathrm e}^{-14 x} x^{2}}{196}-\frac {3 x \,{\mathrm e}^{-14 x}}{1372}-\frac {3 \,{\mathrm e}^{-14 x}}{19208}\right )\) | \(69\) |
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. 42 vs.
\(2 (14) = 28\).
time = 0.27, size = 42, normalized size = 2.21 \begin {gather*} \frac {16384}{343} \, {\left (1372 \, x^{3} + 294 \, x^{2} + 42 \, x + 3\right )} e^{\left (-14 \, x - 3\right )} - \frac {49152}{343} \, {\left (98 \, x^{2} + 14 \, x + 1\right )} e^{\left (-14 \, x - 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 14, normalized size = 0.74 \begin {gather*} x^{3} e^{\left (-14 \, x + 16 \, \log \left (2\right ) - 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 12, normalized size = 0.63 \begin {gather*} \frac {65536 x^{3} e^{- 14 x}}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 14, normalized size = 0.74 \begin {gather*} x^{3} e^{\left (-14 \, x + 16 \, \log \left (2\right ) - 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.29, size = 11, normalized size = 0.58 \begin {gather*} 65536\,x^3\,{\mathrm {e}}^{-14\,x}\,{\mathrm {e}}^{-3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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