Optimal. Leaf size=103 \[ -\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (4+3 x)}{\sqrt {3} \sqrt [3]{28+54 x+27 x^2}}\right )}{3\ 2^{2/3} \sqrt {3}}-\frac {\log (2+3 x)}{6\ 2^{2/3}}+\frac {\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{6\ 2^{2/3}} \]
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Rubi [A]
time = 0.01, 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 = {766}
\begin {gather*} -\frac {\text {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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 766
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 [A]
time = 0.22, size = 171, normalized size = 1.66 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.41, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (2+3 x \right ) \left (27 x^{2}+54 x +28\right )^{\frac {1}{3}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 214 vs.
\(2 (80) = 160\).
time = 6.55, 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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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