Optimal. Leaf size=123 \[ \frac {\tan ^{-1}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d} \]
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Rubi [A] time = 0.02, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {394} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 394
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx &=\frac {\tan ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\tan ^{-1}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 136, normalized size = 1.11 \begin {gather*} -\frac {9 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {3 x^2}{2},-\frac {x^2}{6}\right )}{d \left (x^2+6\right ) \sqrt [3]{3 x^2+2} \left (x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-\frac {3 x^2}{2},-\frac {x^2}{6}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-\frac {3 x^2}{2},-\frac {x^2}{6}\right )\right )-9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {3 x^2}{2},-\frac {x^2}{6}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 4.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 (d x^{2} + 6 \, d\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 58.10, size = 549, normalized size = 4.46 \begin {gather*} -\frac {24 \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \ln \left (-\frac {192 x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{6}-4 x \RootOf \left (\textit {\_Z}^{6}+54\right )^{7}+9 x^{2} \RootOf \left (\textit {\_Z}^{6}+54\right )^{4}-288 \left (3 x^{2}+2\right )^{\frac {1}{3}} x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{4}+6 \left (3 x^{2}+2\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{6}+54\right )^{5}-18 \RootOf \left (\textit {\_Z}^{6}+54\right )^{4}-108 \left (3 x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+54\right )^{2}+324 \left (3 x^{2}+2\right )^{\frac {2}{3}}}{x^{2}+6}\right )-\RootOf \left (\textit {\_Z}^{6}+54\right ) \ln \left (-\frac {768 x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{6}+54\right )^{5}-16 x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{6}-36 x^{2} \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{3}+1152 \left (3 x^{2}+2\right )^{\frac {1}{3}} x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{6}+54\right )^{3}-72 \left (3 x^{2}+2\right )^{\frac {1}{3}} x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{4}+\left (3 x^{2}+2\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{6}+54\right )^{5}+72 \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{3}-432 \left (3 x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )+18 \left (3 x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+54\right )^{2}+54 \left (3 x^{2}+2\right )^{\frac {2}{3}}}{x^{2}+6}\right )}{24 d} \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 (d x^{2} + 6 \, d\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{3}}}\,{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+2\right )}^{1/3}\,\left (d\,x^2+6\,d\right )} \,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*} \frac {\int \frac {1}{x^{2} \sqrt [3]{3 x^{2} + 2} + 6 \sqrt [3]{3 x^{2} + 2}}\, dx}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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