Optimal. Leaf size=18 \[ -1+\frac {1}{x^3}-\frac {8 e^3}{x^2 (3+x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 36, normalized size of antiderivative = 2.00, number of steps
used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {1608, 27, 1834}
\begin {gather*} \frac {1}{x^3}-\frac {8 e^3}{3 x^2}-\frac {8 e^3}{9 (x+3)}+\frac {8 e^3}{9 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1608
Rule 1834
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-27-18 x-3 x^2+e^3 \left (48 x+24 x^2\right )}{x^4 \left (9+6 x+x^2\right )} \, dx\\ &=\int \frac {-27-18 x-3 x^2+e^3 \left (48 x+24 x^2\right )}{x^4 (3+x)^2} \, dx\\ &=\int \left (-\frac {3}{x^4}+\frac {16 e^3}{3 x^3}-\frac {8 e^3}{9 x^2}+\frac {8 e^3}{9 (3+x)^2}\right ) \, dx\\ &=\frac {1}{x^3}-\frac {8 e^3}{3 x^2}+\frac {8 e^3}{9 x}-\frac {8 e^3}{9 (3+x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {3+x-8 e^3 x}{x^3 (3+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 28, normalized size = 1.56
method | result | size |
norman | \(\frac {3+\left (-8 \,{\mathrm e}^{3}+1\right ) x}{\left (3+x \right ) x^{3}}\) | \(20\) |
risch | \(\frac {3+\left (-8 \,{\mathrm e}^{3}+1\right ) x}{\left (3+x \right ) x^{3}}\) | \(20\) |
gosper | \(-\frac {8 x \,{\mathrm e}^{3}-x -3}{x^{3} \left (3+x \right )}\) | \(21\) |
default | \(\frac {1}{x^{3}}-\frac {8 \,{\mathrm e}^{3}}{3 x^{2}}+\frac {8 \,{\mathrm e}^{3}}{9 x}-\frac {8 \,{\mathrm e}^{3}}{9 \left (3+x \right )}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 23, normalized size = 1.28 \begin {gather*} -\frac {x {\left (8 \, e^{3} - 1\right )} - 3}{x^{4} + 3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 23, normalized size = 1.28 \begin {gather*} -\frac {8 \, x e^{3} - x - 3}{x^{4} + 3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 19, normalized size = 1.06 \begin {gather*} - \frac {x \left (-1 + 8 e^{3}\right ) - 3}{x^{4} + 3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 29, normalized size = 1.61 \begin {gather*} -\frac {8 \, e^{3}}{9 \, {\left (x + 3\right )}} + \frac {8 \, x^{2} e^{3} - 24 \, x e^{3} + 9}{9 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.63, size = 23, normalized size = 1.28 \begin {gather*} -\frac {x\,\left (8\,{\mathrm {e}}^3-1\right )-3}{x^4+3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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