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 = 0.13, antiderivative size = 77, normalized size of antiderivative = 1.00, 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 44
Rule 2063
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.59, size = 76, normalized size = 0.99 \[ -\frac {3 \, \sqrt {3} \sqrt {b} x - {\left (3 \, x^{2} - b\right )} \log \left (2 \, \sqrt {3} \sqrt {b} + 3 \, x\right ) + {\left (3 \, x^{2} - b\right )} \log \left (-\sqrt {3} \sqrt {b} + 3 \, x\right ) + 3 \, b}{27 \, {\left (3 \, b x^{2} - b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 55, normalized size = 0.71 \[ \frac {\log \left ({\left | 9 \, \sqrt {3} x + 18 \, \sqrt {b} \right |}\right )}{27 \, b} - \frac {\log \left ({\left | -\sqrt {3} x + \sqrt {b} \right |}\right )}{27 \, b} - \frac {1}{9 \, {\left (\sqrt {3} x - \sqrt {b}\right )} \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 43, normalized size = 0.56 \[ \frac {\ln \left (-\RootOf \left (9 \textit {\_Z}^{3}-9 \textit {\_Z} b +2 \sqrt {3}\, b^{\frac {3}{2}}\right )+x \right )}{27 \RootOf \left (9 \textit {\_Z}^{3}-9 \textit {\_Z} b +2 \sqrt {3}\, b^{\frac {3}{2}}\right )^{2}-9 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{9 \, x^{3} + 2 \, \sqrt {3} b^{\frac {3}{2}} - 9 \, b x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 51, normalized size = 0.66 \[ \frac {2\,\sqrt {3}\,\sqrt {27}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\sqrt {27}}{27}+\frac {2\,\sqrt {27}\,x}{9\,\sqrt {b}}\right )}{243\,b}-\frac {\sqrt {3}}{27\,\sqrt {b}\,\left (x-\frac {\sqrt {3}\,\sqrt {b}}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 60, normalized size = 0.78 \[ - \frac {3 \sqrt {3}}{81 \sqrt {b} x - 27 \sqrt {3} b} + \frac {- \frac {\log {\left (- \frac {\sqrt {3} \sqrt {b}}{3} + x \right )}}{27} + \frac {\log {\left (\frac {2 \sqrt {3} \sqrt {b}}{3} + x \right )}}{27}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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