Optimal. Leaf size=97 \[ \frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{12 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{6 \sqrt [3]{2} \sqrt{3}}-\frac{\log (3 x+2)}{12 \sqrt [3]{2}} \]
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Rubi [A] time = 0.0143885, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {751} \[ \frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{12 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{6 \sqrt [3]{2} \sqrt{3}}-\frac{\log (3 x+2)}{12 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
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Rule 751
Rubi steps
\begin{align*} \int \frac{1}{(2+3 x) \sqrt [3]{4+27 x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{4+27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt{3}}-\frac{\log (2+3 x)}{12 \sqrt [3]{2}}+\frac{\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{12 \sqrt [3]{2}}\\ \end{align*}
Mathematica [C] time = 0.0776071, size = 121, normalized size = 1.25 \[ -\frac{\sqrt [3]{\frac{9 x-2 i \sqrt{3}}{3 x+2}} \sqrt [3]{\frac{9 x+2 i \sqrt{3}}{3 x+2}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )}{2\ 3^{2/3} \sqrt [3]{27 x^2+4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.362, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{2+3\,x}{\frac{1}{\sqrt [3]{27\,{x}^{2}+4}}}}\, dx \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}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 28.3654, size = 571, normalized size = 5.89 \begin{align*} -\frac{1}{36} \, \sqrt{6} 2^{\frac{1}{6}} \arctan \left (\frac{2^{\frac{1}{6}}{\left (4 \, \sqrt{6} 2^{\frac{2}{3}}{\left (27 \, x^{2} + 4\right )}^{\frac{2}{3}}{\left (3 \, x - 2\right )} + \sqrt{6} 2^{\frac{1}{3}}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} + 4 \, \sqrt{6}{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (9 \, x^{2} - 12 \, x + 4\right )}\right )}}{18 \,{\left (9 \, x^{3} - 54 \, x^{2} + 12 \, x - 8\right )}}\right ) - \frac{1}{72} \cdot 2^{\frac{2}{3}} \log \left (\frac{2 \cdot 2^{\frac{2}{3}}{\left (27 \, x^{2} + 4\right )}^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (9 \, x^{2} - 12 \, x + 4\right )} - 2 \,{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 \, x - 2\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac{1}{36} \cdot 2^{\frac{2}{3}} \log \left (\frac{2^{\frac{1}{3}}{\left (3 \, x - 2\right )} + 2 \,{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}}{3 \, x + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} + 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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