Integrand size = 22, antiderivative size = 103 \[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\log (2+3 x)}{6\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{6\ 2^{2/3}} \]
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Time = 0.01 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {766} \[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {\arctan \left (\frac {2^{2/3} (3 x+4)}{\sqrt {3} \sqrt [3]{27 x^2+54 x+28}}+\frac {1}{\sqrt {3}}\right )}{3\ 2^{2/3} \sqrt {3}}+\frac {\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac {\log (3 x+2)}{6\ 2^{2/3}} \]
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Rule 766
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.25 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {4\ 2^{2/3}+3\ 2^{2/3} x+\sqrt [3]{28+54 x+27 x^2}}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )-2 \log \left (4\ 2^{2/3}+3\ 2^{2/3} x-2 \sqrt [3]{28+54 x+27 x^2}\right )+\log \left (16 \sqrt [3]{2}+24 \sqrt [3]{2} x+9 \sqrt [3]{2} x^2+2^{2/3} (4+3 x) \sqrt [3]{28+54 x+27 x^2}+2 \left (28+54 x+27 x^2\right )^{2/3}\right )}{18\ 2^{2/3}} \]
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\[\int \frac {1}{\left (2+3 x \right ) \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (80) = 160\).
Time = 2.00 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.08 \[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=-\frac {1}{18} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} {\left (2 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {2}{3}} {\left (3 \, x + 4\right )} + 4^{\frac {1}{3}} \sqrt {3} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} - 4 \, \sqrt {3} {\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (9 \, x^{2} + 24 \, x + 16\right )}\right )}}{18 \, {\left (9 \, x^{3} + 54 \, x^{2} + 84 \, x + 40\right )}}\right ) - \frac {1}{72} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (9 \, x^{2} + 24 \, x + 16\right )} + 2 \, {\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 4\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac {1}{36} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (3 \, x + 4\right )} - 2 \, {\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}}}{3 \, x + 2}\right ) \]
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\[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \]
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\[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}} \,d x } \]
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\[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=\int { \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx=\int \frac {1}{\left (3\,x+2\right )\,{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \]
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