Optimal. Leaf size=103 \[ \frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt{3}}-\frac{\log (3 x+2)}{6\ 2^{2/3}} \]
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Rubi [A] time = 0.0187359, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {752} \[ \frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt{3}}-\frac{\log (3 x+2)}{6\ 2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 752
Rubi steps
\begin{align*} \int \frac{1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2^{2/3} (4+3 x)}{\sqrt{3} \sqrt [3]{28+54 x+27 x^2}}\right )}{3\ 2^{2/3} \sqrt{3}}-\frac{\log (2+3 x)}{6\ 2^{2/3}}+\frac{\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{6\ 2^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0767898, size = 127, normalized size = 1.23 \[ -\frac{\sqrt [3]{\frac{9 x-i \sqrt{3}+9}{3 x+2}} \sqrt [3]{\frac{9 x+i \sqrt{3}+9}{3 x+2}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )}{2\ 3^{2/3} \sqrt [3]{27 x^2+54 x+28}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.678, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{2+3\,x}{\frac{1}{\sqrt [3]{27\,{x}^{2}+54\,x+28}}}}\, 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} + 54 \, x + 28\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 = 25.4418, size = 626, normalized size = 6.08 \begin{align*} -\frac{1}{18} \cdot 4^{\frac{1}{6}} \sqrt{3} \arctan \left (\frac{4^{\frac{1}{6}}{\left (2 \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{2}{3}}{\left (3 \, x + 4\right )} + 4^{\frac{1}{3}} \sqrt{3}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} - 4 \, \sqrt{3}{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}{\left (9 \, x^{2} + 24 \, x + 16\right )}\right )}}{18 \,{\left (9 \, x^{3} + 54 \, x^{2} + 84 \, x + 40\right )}}\right ) - \frac{1}{72} \cdot 4^{\frac{2}{3}} \log \left (\frac{4^{\frac{2}{3}}{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (9 \, x^{2} + 24 \, x + 16\right )} + 2 \,{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}{\left (3 \, x + 4\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac{1}{36} \cdot 4^{\frac{2}{3}} \log \left (\frac{4^{\frac{1}{3}}{\left (3 \, x + 4\right )} - 2 \,{\left (27 \, x^{2} + 54 \, x + 28\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} + 54 x + 28}}\, 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} + 54 \, x + 28\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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