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 = 0.02, antiderivative size = 103, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}
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*}
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Mathematica [C] time = 0.08, size = 127, normalized size = 1.23 \begin {gather*} -\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}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.22, size = 201, normalized size = 1.95 \begin {gather*} \frac {\log \left (-2 \sqrt [3]{27 x^2+54 x+28}+3\ 2^{2/3} x+4\ 2^{2/3}\right )}{9\ 2^{2/3}}-\frac {\log \left (9 \sqrt [3]{2} x^2+2 \left (27 x^2+54 x+28\right )^{2/3}+\left (3\ 2^{2/3} x+4\ 2^{2/3}\right ) \sqrt [3]{27 x^2+54 x+28}+24 \sqrt [3]{2} x+16 \sqrt [3]{2}\right )}{18\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{27 x^2+54 x+28}}{\sqrt {3}}+2^{2/3} \sqrt {3} x+\frac {4\ 2^{2/3}}{\sqrt {3}}}{\sqrt [3]{27 x^2+54 x+28}}\right )}{3\ 2^{2/3} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.86, size = 214, normalized size = 2.08 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.72, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x +2\right ) \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}\, dx \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (3\,x+2\right )\,{\left (27\,x^2+54\,x+28\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} + 54 x + 28}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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