Optimal. Leaf size=18 \[ -\frac {6-x}{2 \sqrt {4+x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {651}
\begin {gather*} -\frac {6-x}{2 \sqrt {x^2+4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin {align*} \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx &=-\frac {6-x}{2 \sqrt {4+x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 16, normalized size = 0.89 \begin {gather*} \frac {-6+x}{2 \sqrt {4+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 21, normalized size = 1.17
method | result | size |
gosper | \(\frac {x -6}{2 \sqrt {x^{2}+4}}\) | \(13\) |
trager | \(\frac {x -6}{2 \sqrt {x^{2}+4}}\) | \(13\) |
risch | \(\frac {x -6}{2 \sqrt {x^{2}+4}}\) | \(13\) |
default | \(\frac {x}{2 \sqrt {x^{2}+4}}-\frac {3}{\sqrt {x^{2}+4}}\) | \(21\) |
meijerg | \(\frac {x}{4 \sqrt {1+\frac {x^{2}}{4}}}+\frac {\frac {3 \sqrt {\pi }}{2}-\frac {3 \sqrt {\pi }}{2 \sqrt {1+\frac {x^{2}}{4}}}}{\sqrt {\pi }}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 20, normalized size = 1.11 \begin {gather*} \frac {x}{2 \, \sqrt {x^{2} + 4}} - \frac {3}{\sqrt {x^{2} + 4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (12) = 24\).
time = 2.79, size = 25, normalized size = 1.39 \begin {gather*} \frac {x^{2} + \sqrt {x^{2} + 4} {\left (x - 6\right )} + 4}{2 \, {\left (x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.70, size = 20, normalized size = 1.11 \begin {gather*} \frac {x}{2 \sqrt {x^{2} + 4}} - \frac {3}{\sqrt {x^{2} + 4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.30, size = 12, normalized size = 0.67 \begin {gather*} \frac {x - 6}{2 \, \sqrt {x^{2} + 4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 12, normalized size = 0.67 \begin {gather*} \frac {x-6}{2\,\sqrt {x^2+4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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