Optimal. Leaf size=27 \[ -x+\frac {1}{256} (4-e)^2 x^2 \left (-e^{-5+x}+x\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 47, normalized size of antiderivative = 1.74, number of steps used = 18, number of rules used = 5, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {6, 12, 2196, 2176, 2194} \begin {gather*} \frac {1}{256} (4-e)^2 x^3-\frac {1}{16} e^{x-5} x^2+\frac {1}{256} (8-e) e^{x-4} x^2-x \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{256} \left (-256+(48-24 e) x^2+3 e^2 x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx\\ &=\int \frac {1}{256} \left (-256+\left (48-24 e+3 e^2\right ) x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx\\ &=\frac {1}{256} \int \left (-256+\left (48-24 e+3 e^2\right ) x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx\\ &=-x+\frac {1}{256} (4-e)^2 x^3+\frac {1}{256} \int e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right ) \, dx\\ &=-x+\frac {1}{256} (4-e)^2 x^3+\frac {1}{256} \int \left (-32 e^{-5+x} x-16 e^{-5+x} x^2+8 \left (1-\frac {e}{8}\right ) e^{-4+x} x (2+x)\right ) \, dx\\ &=-x+\frac {1}{256} (4-e)^2 x^3-\frac {1}{16} \int e^{-5+x} x^2 \, dx-\frac {1}{8} \int e^{-5+x} x \, dx+\frac {1}{256} (8-e) \int e^{-4+x} x (2+x) \, dx\\ &=-x-\frac {1}{8} e^{-5+x} x-\frac {1}{16} e^{-5+x} x^2+\frac {1}{256} (4-e)^2 x^3+\frac {1}{8} \int e^{-5+x} \, dx+\frac {1}{8} \int e^{-5+x} x \, dx+\frac {1}{256} (8-e) \int \left (2 e^{-4+x} x+e^{-4+x} x^2\right ) \, dx\\ &=\frac {e^{-5+x}}{8}-x-\frac {1}{16} e^{-5+x} x^2+\frac {1}{256} (4-e)^2 x^3-\frac {1}{8} \int e^{-5+x} \, dx+\frac {1}{256} (8-e) \int e^{-4+x} x^2 \, dx+\frac {1}{128} (8-e) \int e^{-4+x} x \, dx\\ &=-x+\frac {1}{128} (8-e) e^{-4+x} x-\frac {1}{16} e^{-5+x} x^2+\frac {1}{256} (8-e) e^{-4+x} x^2+\frac {1}{256} (4-e)^2 x^3+\frac {1}{128} (-8+e) \int e^{-4+x} \, dx+\frac {1}{128} (-8+e) \int e^{-4+x} x \, dx\\ &=-\frac {1}{128} (8-e) e^{-4+x}-x-\frac {1}{16} e^{-5+x} x^2+\frac {1}{256} (8-e) e^{-4+x} x^2+\frac {1}{256} (4-e)^2 x^3+\frac {1}{128} (8-e) \int e^{-4+x} \, dx\\ &=-x-\frac {1}{16} e^{-5+x} x^2+\frac {1}{256} (8-e) e^{-4+x} x^2+\frac {1}{256} (4-e)^2 x^3\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.03, size = 65, normalized size = 2.41 \begin {gather*} \frac {-256 e^5 x-16 e^x x^2+8 e^{1+x} x^2-e^{2+x} x^2+16 e^5 x^3-8 e^6 x^3+e^7 x^3}{256 e^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 48, normalized size = 1.78 \begin {gather*} \frac {1}{256} \, x^{3} e^{2} - \frac {1}{32} \, x^{3} e + \frac {1}{16} \, x^{3} - \frac {1}{256} \, {\left (x^{2} e^{2} - 8 \, x^{2} e + 16 \, x^{2}\right )} e^{\left (x - 5\right )} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 50, normalized size = 1.85 \begin {gather*} \frac {1}{256} \, x^{3} e^{2} - \frac {1}{32} \, x^{3} e + \frac {1}{16} \, x^{3} - \frac {1}{256} \, x^{2} e^{\left (x - 3\right )} + \frac {1}{32} \, x^{2} e^{\left (x - 4\right )} - \frac {1}{16} \, x^{2} e^{\left (x - 5\right )} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 41, normalized size = 1.52
method | result | size |
norman | \(\left (\frac {{\mathrm e}^{2}}{256}-\frac {{\mathrm e}}{32}+\frac {1}{16}\right ) x^{3}+\left (-\frac {{\mathrm e}^{2}}{256}+\frac {{\mathrm e}}{32}-\frac {1}{16}\right ) x^{2} {\mathrm e}^{x -5}-x\) | \(41\) |
risch | \(-\frac {\left ({\mathrm e}^{2}-8 \,{\mathrm e}+16\right ) x^{2} {\mathrm e}^{x -5}}{256}+\frac {x^{3} {\mathrm e}^{2}}{256}-\frac {x^{3} {\mathrm e}}{32}+\frac {x^{3}}{16}-x\) | \(41\) |
default | \(-x +\frac {x^{3}}{16}-\frac {x^{3} {\mathrm e}}{32}+\frac {x^{3} {\mathrm e}^{2}}{256}-\frac {5 \,{\mathrm e}^{x -5} \left (x -5\right )}{8}-\frac {25 \,{\mathrm e}^{x -5}}{16}-\frac {{\mathrm e}^{x -5} \left (x -5\right )^{2}}{16}+\frac {35 \,{\mathrm e} \,{\mathrm e}^{x -5}}{32}-\frac {35 \,{\mathrm e}^{2} {\mathrm e}^{x -5}}{256}+\frac {3 \,{\mathrm e} \left ({\mathrm e}^{x -5} \left (x -5\right )-{\mathrm e}^{x -5}\right )}{8}+\frac {{\mathrm e} \left ({\mathrm e}^{x -5} \left (x -5\right )^{2}-2 \,{\mathrm e}^{x -5} \left (x -5\right )+2 \,{\mathrm e}^{x -5}\right )}{32}-\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}^{x -5} \left (x -5\right )-{\mathrm e}^{x -5}\right )}{64}-\frac {{\mathrm e}^{2} \left ({\mathrm e}^{x -5} \left (x -5\right )^{2}-2 \,{\mathrm e}^{x -5} \left (x -5\right )+2 \,{\mathrm e}^{x -5}\right )}{256}\) | \(172\) |
derivativedivides | \(-x +5+\frac {x^{3}}{16}-\frac {x^{3} {\mathrm e}}{32}+\frac {x^{3} {\mathrm e}^{2}}{256}-\frac {5 \,{\mathrm e}^{x -5} \left (x -5\right )}{8}-\frac {25 \,{\mathrm e}^{x -5}}{16}-\frac {{\mathrm e}^{x -5} \left (x -5\right )^{2}}{16}+\frac {35 \,{\mathrm e} \,{\mathrm e}^{x -5}}{32}-\frac {35 \,{\mathrm e}^{2} {\mathrm e}^{x -5}}{256}+\frac {3 \,{\mathrm e} \left ({\mathrm e}^{x -5} \left (x -5\right )-{\mathrm e}^{x -5}\right )}{8}+\frac {{\mathrm e} \left ({\mathrm e}^{x -5} \left (x -5\right )^{2}-2 \,{\mathrm e}^{x -5} \left (x -5\right )+2 \,{\mathrm e}^{x -5}\right )}{32}-\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}^{x -5} \left (x -5\right )-{\mathrm e}^{x -5}\right )}{64}-\frac {{\mathrm e}^{2} \left ({\mathrm e}^{x -5} \left (x -5\right )^{2}-2 \,{\mathrm e}^{x -5} \left (x -5\right )+2 \,{\mathrm e}^{x -5}\right )}{256}\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 40, normalized size = 1.48 \begin {gather*} \frac {1}{256} \, x^{3} e^{2} - \frac {1}{32} \, x^{3} e - \frac {1}{256} \, x^{2} {\left (e^{2} - 8 \, e + 16\right )} e^{\left (x - 5\right )} + \frac {1}{16} \, x^{3} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.08, size = 30, normalized size = 1.11 \begin {gather*} \frac {x^3\,{\left (\mathrm {e}-4\right )}^2}{256}-x-\frac {x^2\,{\mathrm {e}}^{x-5}\,{\left (\mathrm {e}-4\right )}^2}{256} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 44, normalized size = 1.63 \begin {gather*} x^{3} \left (- \frac {e}{32} + \frac {e^{2}}{256} + \frac {1}{16}\right ) - x + \frac {\left (- 16 x^{2} - x^{2} e^{2} + 8 e x^{2}\right ) e^{x - 5}}{256} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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