Optimal. Leaf size=77 \[ \frac{1}{3 \sqrt{3} \sqrt{b} \left (\sqrt{3} \sqrt{b}-3 x\right )}-\frac{\log \left (\sqrt{b}-\sqrt{3} x\right )}{27 b}+\frac{\log \left (2 \sqrt{b}+\sqrt{3} x\right )}{27 b} \]
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Rubi [A] time = 0.13389, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2063, 44} \[ \frac{1}{3 \sqrt{3} \sqrt{b} \left (\sqrt{3} \sqrt{b}-3 x\right )}-\frac{\log \left (\sqrt{b}-\sqrt{3} x\right )}{27 b}+\frac{\log \left (2 \sqrt{b}+\sqrt{3} x\right )}{27 b} \]
Antiderivative was successfully verified.
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Rule 2063
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{2 \sqrt{3} b^{3/2}-9 b x+9 x^3} \, dx &=\left (324 b^3\right ) \int \frac{1}{\left (6 \sqrt{3} b^{3/2}-18 b x\right )^2 \left (6 \sqrt{3} b^{3/2}+9 b x\right )} \, dx\\ &=\left (324 b^3\right ) \int \left (\frac{1}{324 \sqrt{3} b^{7/2} \left (\sqrt{3} \sqrt{b}-3 x\right )^2}+\frac{1}{2916 b^4 \left (\sqrt{3} \sqrt{b}-3 x\right )}+\frac{1}{2916 b^4 \left (2 \sqrt{3} \sqrt{b}+3 x\right )}\right ) \, dx\\ &=\frac{1}{3 \sqrt{3} \sqrt{b} \left (\sqrt{3} \sqrt{b}-3 x\right )}-\frac{\log \left (\sqrt{b}-\sqrt{3} x\right )}{27 b}+\frac{\log \left (2 \sqrt{b}+\sqrt{3} x\right )}{27 b}\\ \end{align*}
Mathematica [A] time = 0.0484474, size = 143, normalized size = 1.86 \[ -\frac{\left (3 x-\sqrt{3} \sqrt{b}\right ) \left (2 \sqrt{3} \sqrt{b}+3 x\right ) \left (\left (3 x-\sqrt{3} \sqrt{b}\right ) \log \left (3 x-\sqrt{3} \sqrt{b}\right )+\left (\sqrt{3} \sqrt{b}-3 x\right ) \log \left (2 \sqrt{3} \sqrt{b}+3 x\right )+3 \sqrt{3} \sqrt{b}\right )}{81 b \left (2 \sqrt{3} b^{3/2}-9 b x+9 x^3\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 43, normalized size = 0.6 \begin{align*}{\frac{1}{9}\sum _{{\it \_R}={\it RootOf} \left ( -9\,b{\it \_Z}+9\,{{\it \_Z}}^{3}+2\,{b}^{3/2}\sqrt{3} \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{3\,{{\it \_R}}^{2}-b}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{9 \, x^{3} + 2 \, \sqrt{3} b^{\frac{3}{2}} - 9 \, b x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38785, size = 184, normalized size = 2.39 \begin{align*} -\frac{3 \, \sqrt{3} \sqrt{b} x -{\left (3 \, x^{2} - b\right )} \log \left (2 \, \sqrt{3} \sqrt{b} + 3 \, x\right ) +{\left (3 \, x^{2} - b\right )} \log \left (-\sqrt{3} \sqrt{b} + 3 \, x\right ) + 3 \, b}{27 \,{\left (3 \, b x^{2} - b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.443516, size = 60, normalized size = 0.78 \begin{align*} - \frac{3 \sqrt{3}}{81 \sqrt{b} x - 27 \sqrt{3} b} + \frac{- \frac{\log{\left (- \frac{\sqrt{3} \sqrt{b}}{3} + x \right )}}{27} + \frac{\log{\left (\frac{2 \sqrt{3} \sqrt{b}}{3} + x \right )}}{27}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09659, size = 73, normalized size = 0.95 \begin{align*} \frac{\log \left ({\left | \sqrt{3} x + 2 \, \sqrt{b} \right |}\right )}{27 \, b} - \frac{\log \left ({\left | -\sqrt{3} x + \sqrt{b} \right |}\right )}{27 \, b} - \frac{1}{9 \,{\left (\sqrt{3} x - \sqrt{b}\right )} \sqrt{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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