Optimal. Leaf size=26 \[ \frac {x}{18 \sqrt {2} \sqrt {3-x} \sqrt {3+x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {39}
\begin {gather*} \frac {x}{18 \sqrt {2} \sqrt {3-x} \sqrt {x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 39
Rubi steps
\begin {align*} \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx &=\frac {x}{18 \sqrt {2} \sqrt {3-x} \sqrt {3+x}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 21, normalized size = 0.81 \begin {gather*} \frac {x}{18 \sqrt {6-2 x} \sqrt {3+x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 30, normalized size = 1.15
method | result | size |
gosper | \(-\frac {\left (-3+x \right ) x}{9 \sqrt {3+x}\, \left (6-2 x \right )^{\frac {3}{2}}}\) | \(19\) |
default | \(\frac {1}{6 \sqrt {6-2 x}\, \sqrt {3+x}}-\frac {\sqrt {6-2 x}}{36 \sqrt {3+x}}\) | \(30\) |
risch | \(\frac {\sqrt {\left (3+x \right ) \left (6-2 x \right )}\, \sqrt {2}\, x}{36 \sqrt {3+x}\, \sqrt {6-2 x}\, \sqrt {-\left (-3+x \right ) \left (3+x \right )}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 12, normalized size = 0.46 \begin {gather*} \frac {x}{18 \, \sqrt {-2 \, x^{2} + 18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.53, size = 22, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {x + 3} x \sqrt {-2 \, x + 6}}{36 \, {\left (x^{2} - 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 8.37, size = 92, normalized size = 3.54 \begin {gather*} \begin {cases} - \frac {\sqrt {2} \sqrt {-1 + \frac {6}{x + 3}} \left (x + 3\right )}{36 x - 108} + \frac {3 \sqrt {2} \sqrt {-1 + \frac {6}{x + 3}}}{36 x - 108} & \text {for}\: \frac {1}{\left |{x + 3}\right |} > \frac {1}{6} \\- \frac {\sqrt {2} i}{36 \sqrt {1 - \frac {6}{x + 3}}} + \frac {\sqrt {2} i}{12 \sqrt {1 - \frac {6}{x + 3}} \left (x + 3\right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (18) = 36\).
time = 1.25, size = 71, normalized size = 2.73 \begin {gather*} \frac {\sqrt {2} {\left (\sqrt {6} - \sqrt {-x + 3}\right )}}{144 \, \sqrt {x + 3}} - \frac {\sqrt {2} \sqrt {x + 3} \sqrt {-x + 3}}{72 \, {\left (x - 3\right )}} - \frac {\sqrt {2} \sqrt {x + 3}}{144 \, {\left (\sqrt {6} - \sqrt {-x + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 22, normalized size = 0.85 \begin {gather*} -\frac {x\,\sqrt {6-2\,x}}{\left (36\,x-108\right )\,\sqrt {x+3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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