Optimal. Leaf size=25 \[ e^{3+x-\frac {4 \left (-2+x^2\right )}{(3+x)^2}}-x+5 \log (5) \]
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Rubi [F]
time = 0.63, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-27-27 x-9 x^2-x^3+e^{\frac {35+27 x+5 x^2+x^3}{9+6 x+x^2}} \left (11+3 x+9 x^2+x^3\right )}{27+27 x+9 x^2+x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {27}{(3+x)^3}-\frac {27 x}{(3+x)^3}-\frac {9 x^2}{(3+x)^3}-\frac {x^3}{(3+x)^3}+\frac {e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}} \left (11+3 x+9 x^2+x^3\right )}{(3+x)^3}\right ) \, dx\\ &=\frac {27}{2 (3+x)^2}-9 \int \frac {x^2}{(3+x)^3} \, dx-27 \int \frac {x}{(3+x)^3} \, dx-\int \frac {x^3}{(3+x)^3} \, dx+\int \frac {e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}} \left (11+3 x+9 x^2+x^3\right )}{(3+x)^3} \, dx\\ &=\frac {27}{2 (3+x)^2}-\frac {9 x^2}{2 (3+x)^2}-9 \int \left (\frac {9}{(3+x)^3}-\frac {6}{(3+x)^2}+\frac {1}{3+x}\right ) \, dx+\int \left (e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}}+\frac {56 e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}}}{(3+x)^3}-\frac {24 e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}}}{(3+x)^2}\right ) \, dx-\int \left (1-\frac {27}{(3+x)^3}+\frac {27}{(3+x)^2}-\frac {9}{3+x}\right ) \, dx\\ &=-x+\frac {81}{2 (3+x)^2}-\frac {9 x^2}{2 (3+x)^2}-\frac {27}{3+x}-24 \int \frac {e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}}}{(3+x)^2} \, dx+56 \int \frac {e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}}}{(3+x)^3} \, dx+\int e^{\frac {35+27 x+5 x^2+x^3}{(3+x)^2}} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.14, size = 23, normalized size = 0.92 \begin {gather*} e^{-1+x-\frac {28}{(3+x)^2}+\frac {24}{3+x}}-x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 25, normalized size = 1.00
method | result | size |
risch | \(-x +{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{\left (3+x \right )^{2}}}\) | \(25\) |
norman | \(\frac {27 x +{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}} x^{2}-x^{3}+6 \,{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}} x +9 \,{\mathrm e}^{\frac {x^{3}+5 x^{2}+27 x +35}{x^{2}+6 x +9}}+54}{\left (3+x \right )^{2}}\) | \(101\) |
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. 90 vs.
\(2 (24) = 48\).
time = 0.42, size = 90, normalized size = 3.60 \begin {gather*} -x - \frac {27 \, {\left (4 \, x + 9\right )}}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {27 \, {\left (2 \, x + 5\right )}}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {27 \, {\left (2 \, x + 3\right )}}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {27}{2 \, {\left (x^{2} + 6 \, x + 9\right )}} + e^{\left (x - \frac {28}{x^{2} + 6 \, x + 9} + \frac {24}{x + 3} - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 29, normalized size = 1.16 \begin {gather*} -x + e^{\left (\frac {x^{3} + 5 \, x^{2} + 27 \, x + 35}{x^{2} + 6 \, x + 9}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 24, normalized size = 0.96 \begin {gather*} - x + e^{\frac {x^{3} + 5 x^{2} + 27 x + 35}{x^{2} + 6 x + 9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 33, normalized size = 1.32 \begin {gather*} -x + e^{\left (\frac {9 \, x^{3} + 10 \, x^{2} + 33 \, x}{9 \, {\left (x^{2} + 6 \, x + 9\right )}} + \frac {35}{9}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 63, normalized size = 2.52 \begin {gather*} {\mathrm {e}}^{\frac {x^3}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {5\,x^2}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {35}{x^2+6\,x+9}}\,{\mathrm {e}}^{\frac {27\,x}{x^2+6\,x+9}}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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