Optimal. Leaf size=24 \[ \frac {x}{9 \sqrt {3-b x} \sqrt {b x+3}} \]
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Rubi [A] time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {39} \[ \frac {x}{9 \sqrt {3-b x} \sqrt {b x+3}} \]
Antiderivative was successfully verified.
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Rule 39
Rubi steps
\begin {align*} \int \frac {1}{(3-b x)^{3/2} (3+b x)^{3/2}} \, dx &=\frac {x}{9 \sqrt {3-b x} \sqrt {3+b x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 0.79 \[ \frac {x}{9 \sqrt {9-b^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 29, normalized size = 1.21 \[ -\frac {\sqrt {b x + 3} \sqrt {-b x + 3} x}{9 \, {\left (b^{2} x^{2} - 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.10, size = 82, normalized size = 3.42 \[ \frac {\sqrt {6} - \sqrt {-b x + 3}}{36 \, \sqrt {b x + 3} b} - \frac {\sqrt {b x + 3} \sqrt {-b x + 3}}{18 \, {\left (b x - 3\right )} b} - \frac {\sqrt {b x + 3}}{36 \, b {\left (\sqrt {6} - \sqrt {-b x + 3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.79 \[ \frac {x}{9 \sqrt {-b x +3}\, \sqrt {b x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 15, normalized size = 0.62 \[ \frac {x}{9 \, \sqrt {-b^{2} x^{2} + 9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 26, normalized size = 1.08 \[ -\frac {x\,\sqrt {3-b\,x}}{\sqrt {b\,x+3}\,\left (9\,b\,x-27\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.16, size = 73, normalized size = 3.04 \[ - \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & \frac {1}{2}, \frac {3}{2}, 2 \\\frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 2 & 0 \end {matrix} \middle | {\frac {9}{b^{2} x^{2}}} \right )}}{18 \pi ^{\frac {3}{2}} b} + \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1 & \\\frac {1}{4}, \frac {3}{4} & - \frac {1}{2}, 0, 1, 0 \end {matrix} \middle | {\frac {9 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{18 \pi ^{\frac {3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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