Optimal. Leaf size=26 \[ \frac {x}{18 \sqrt {2} \sqrt {3-x} \sqrt {x+3}} \]
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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 = 0.01, size = 21, normalized size = 0.81 \begin {gather*} \frac {x}{18 \sqrt {6-2 x} \sqrt {x+3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.29, 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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giac [B] time = 1.04, 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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maple [A] time = 0.00, size = 19, normalized size = 0.73 \begin {gather*} -\frac {\left (x -3\right ) x}{9 \sqrt {x +3}\, \left (-2 x +6\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, 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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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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sympy [A] time = 20.45, size = 90, normalized size = 3.46 \begin {gather*} \begin {cases} \frac {\sqrt {2}}{36 \sqrt {-1 + \frac {6}{x + 3}}} - \frac {\sqrt {2}}{12 \sqrt {-1 + \frac {6}{x + 3}} \left (x + 3\right )} & \text {for}\: \frac {6}{\left |{x + 3}\right |} > 1 \\\frac {\sqrt {2} i \sqrt {1 - \frac {6}{x + 3}} \left (x + 3\right )}{108 - 36 x} - \frac {3 \sqrt {2} i \sqrt {1 - \frac {6}{x + 3}}}{108 - 36 x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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