Optimal. Leaf size=24 \[ \frac{x}{9 \sqrt{3-b x} \sqrt{b x+3}} \]
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Rubi [A] time = 0.0025522, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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*}
Mathematica [A] time = 0.007636, size = 19, normalized size = 0.79 \[ \frac{x}{9 \sqrt{9-b^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 0.8 \begin{align*}{\frac{x}{9}{\frac{1}{\sqrt{-bx+3}}}{\frac{1}{\sqrt{bx+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995133, size = 20, normalized size = 0.83 \begin{align*} \frac{x}{9 \, \sqrt{-b^{2} x^{2} + 9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55026, size = 69, normalized size = 2.88 \begin{align*} -\frac{\sqrt{b x + 3} \sqrt{-b x + 3} x}{9 \,{\left (b^{2} x^{2} - 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.32864, size = 73, normalized size = 3.04 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.06276, size = 111, normalized size = 4.62 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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