Optimal. Leaf size=23 \[ \frac {e^x}{-1-\frac {3 (-4-x) \left (x+x^2\right )}{x}} \]
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Rubi [A]
time = 0.13, antiderivative size = 16, normalized size of antiderivative = 0.70, number of steps
used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6820, 2327}
\begin {gather*} \frac {e^x}{3 x^2+15 x+11} \end {gather*}
Antiderivative was successfully verified.
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Rule 2327
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-4+9 x+3 x^2\right )}{\left (11+15 x+3 x^2\right )^2} \, dx\\ &=\frac {e^x}{11+15 x+3 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 16, normalized size = 0.70 \begin {gather*} \frac {e^x}{11+15 x+3 x^2} \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. \(64\) vs.
\(2(22)=44\).
time = 0.19, size = 65, normalized size = 2.83
method | result | size |
gosper | \(\frac {{\mathrm e}^{x}}{3 x^{2}+15 x +11}\) | \(16\) |
norman | \(\frac {{\mathrm e}^{x}}{3 x^{2}+15 x +11}\) | \(16\) |
risch | \(\frac {{\mathrm e}^{x}}{3 x^{2}+15 x +11}\) | \(16\) |
default | \(\frac {4 \,{\mathrm e}^{x} \left (5+2 x \right )}{31 \left (3 x^{2}+15 x +11\right )}+\frac {3 \,{\mathrm e}^{x} \left (15 x +22\right )}{31 \left (3 x^{2}+15 x +11\right )}-\frac {{\mathrm e}^{x} \left (53 x +55\right )}{31 \left (3 x^{2}+15 x +11\right )}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 15, normalized size = 0.65 \begin {gather*} \frac {e^{x}}{3 \, x^{2} + 15 \, x + 11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 15, normalized size = 0.65 \begin {gather*} \frac {e^{x}}{3 \, x^{2} + 15 \, x + 11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 12, normalized size = 0.52 \begin {gather*} \frac {e^{x}}{3 x^{2} + 15 x + 11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 15, normalized size = 0.65 \begin {gather*} \frac {e^{x}}{3 \, x^{2} + 15 \, x + 11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 15, normalized size = 0.65 \begin {gather*} \frac {{\mathrm {e}}^x}{3\,\left (x^2+5\,x+\frac {11}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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