Optimal. Leaf size=32 \[ \frac {\left (-4+x-e^{5-x} x-x^2\right ) \left (-4+(4+x)^2\right )}{3 x} \]
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Rubi [A] time = 0.09, antiderivative size = 60, normalized size of antiderivative = 1.88, number of steps used = 13, number of rules used = 5, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {12, 14, 2196, 2194, 2176} \begin {gather*} -\frac {x^3}{3}-\frac {1}{3} e^{5-x} x^2-\frac {7 x^2}{3}-\frac {8}{3} e^{5-x} x-\frac {8 x}{3}-4 e^{5-x}-\frac {16}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {48-8 x^2-14 x^3-3 x^4+e^{5-x} \left (4 x^2+6 x^3+x^4\right )}{x^2} \, dx\\ &=\frac {1}{3} \int \left (e^{5-x} \left (4+6 x+x^2\right )+\frac {48-8 x^2-14 x^3-3 x^4}{x^2}\right ) \, dx\\ &=\frac {1}{3} \int e^{5-x} \left (4+6 x+x^2\right ) \, dx+\frac {1}{3} \int \frac {48-8 x^2-14 x^3-3 x^4}{x^2} \, dx\\ &=\frac {1}{3} \int \left (-8+\frac {48}{x^2}-14 x-3 x^2\right ) \, dx+\frac {1}{3} \int \left (4 e^{5-x}+6 e^{5-x} x+e^{5-x} x^2\right ) \, dx\\ &=-\frac {16}{x}-\frac {8 x}{3}-\frac {7 x^2}{3}-\frac {x^3}{3}+\frac {1}{3} \int e^{5-x} x^2 \, dx+\frac {4}{3} \int e^{5-x} \, dx+2 \int e^{5-x} x \, dx\\ &=-\frac {4 e^{5-x}}{3}-\frac {16}{x}-\frac {8 x}{3}-2 e^{5-x} x-\frac {7 x^2}{3}-\frac {1}{3} e^{5-x} x^2-\frac {x^3}{3}+\frac {2}{3} \int e^{5-x} x \, dx+2 \int e^{5-x} \, dx\\ &=-\frac {10 e^{5-x}}{3}-\frac {16}{x}-\frac {8 x}{3}-\frac {8}{3} e^{5-x} x-\frac {7 x^2}{3}-\frac {1}{3} e^{5-x} x^2-\frac {x^3}{3}+\frac {2}{3} \int e^{5-x} \, dx\\ &=-4 e^{5-x}-\frac {16}{x}-\frac {8 x}{3}-\frac {8}{3} e^{5-x} x-\frac {7 x^2}{3}-\frac {1}{3} e^{5-x} x^2-\frac {x^3}{3}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 49, normalized size = 1.53 \begin {gather*} \frac {1}{3} \left (-\frac {48}{x}-8 x-7 x^2-x^3+e^{-x} \left (-12 e^5-8 e^5 x-e^5 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 39, normalized size = 1.22 \begin {gather*} -\frac {x^{4} + 7 \, x^{3} + 8 \, x^{2} + {\left (x^{3} + 8 \, x^{2} + 12 \, x\right )} e^{\left (-x + 5\right )} + 48}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 50, normalized size = 1.56 \begin {gather*} -\frac {x^{4} + x^{3} e^{\left (-x + 5\right )} + 7 \, x^{3} + 8 \, x^{2} e^{\left (-x + 5\right )} + 8 \, x^{2} + 12 \, x e^{\left (-x + 5\right )} + 48}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 38, normalized size = 1.19
method | result | size |
risch | \(-\frac {x^{3}}{3}-\frac {7 x^{2}}{3}-\frac {8 x}{3}-\frac {16}{x}+\frac {\left (-x^{2}-8 x -12\right ) {\mathrm e}^{5-x}}{3}\) | \(38\) |
norman | \(\frac {-16-\frac {8 x^{2}}{3}-\frac {7 x^{3}}{3}-\frac {x^{4}}{3}-4 x \,{\mathrm e}^{5-x}-\frac {8 \,{\mathrm e}^{5-x} x^{2}}{3}-\frac {{\mathrm e}^{5-x} x^{3}}{3}}{x}\) | \(53\) |
derivativedivides | \(-\frac {{\mathrm e}^{5-x} \left (5-x \right )^{2}}{3}+6 \,{\mathrm e}^{5-x} \left (5-x \right )-\frac {77 \,{\mathrm e}^{5-x}}{3}-\frac {16}{x}+255-51 x -\frac {22 \left (5-x \right )^{2}}{3}+\frac {\left (5-x \right )^{3}}{3}\) | \(65\) |
default | \(-\frac {{\mathrm e}^{5-x} \left (5-x \right )^{2}}{3}+6 \,{\mathrm e}^{5-x} \left (5-x \right )-\frac {77 \,{\mathrm e}^{5-x}}{3}-\frac {16}{x}+255-51 x -\frac {22 \left (5-x \right )^{2}}{3}+\frac {\left (5-x \right )^{3}}{3}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 62, normalized size = 1.94 \begin {gather*} -\frac {1}{3} \, x^{3} - \frac {7}{3} \, x^{2} - \frac {1}{3} \, {\left (x^{2} e^{5} + 2 \, x e^{5} + 2 \, e^{5}\right )} e^{\left (-x\right )} - 2 \, {\left (x e^{5} + e^{5}\right )} e^{\left (-x\right )} - \frac {8}{3} \, x - \frac {16}{x} - \frac {4}{3} \, e^{\left (-x + 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 47, normalized size = 1.47 \begin {gather*} -\frac {8\,x}{3}-4\,{\mathrm {e}}^{5-x}-\frac {8\,x\,{\mathrm {e}}^{5-x}}{3}-\frac {x^2\,{\mathrm {e}}^{5-x}}{3}-\frac {16}{x}-\frac {7\,x^2}{3}-\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 36, normalized size = 1.12 \begin {gather*} - \frac {x^{3}}{3} - \frac {7 x^{2}}{3} - \frac {8 x}{3} + \frac {\left (- x^{2} - 8 x - 12\right ) e^{5 - x}}{3} - \frac {16}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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