Optimal. Leaf size=151 \[ \frac {\sqrt {b} \tan ^{-1}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{b x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt {b} x}\right )}{12\ 2^{5/6} d}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt [6]{2} \sqrt {3} \left (\sqrt [3]{2}-\sqrt [3]{b x^2+2}\right )}{\sqrt {b} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{3 \sqrt {2}}\right )}{12\ 2^{5/6} d} \]
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Rubi [A] time = 0.03, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {394} \begin {gather*} \frac {\sqrt {b} \tan ^{-1}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{b x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt {b} x}\right )}{12\ 2^{5/6} d}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt [6]{2} \sqrt {3} \left (\sqrt [3]{2}-\sqrt [3]{b x^2+2}\right )}{\sqrt {b} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{3 \sqrt {2}}\right )}{12\ 2^{5/6} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 394
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{2+b x^2} \left (\frac {18 d}{b}+d x^2\right )} \, dx &=\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{3 \sqrt {2}}\right )}{12\ 2^{5/6} d}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\left (\sqrt [3]{2}-\sqrt [3]{2+b x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {b} x}\right )}{12\ 2^{5/6} d}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt [6]{2} \sqrt {3} \left (\sqrt [3]{2}-\sqrt [3]{2+b x^2}\right )}{\sqrt {b} x}\right )}{4\ 2^{5/6} \sqrt {3} d}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 148, normalized size = 0.98 \begin {gather*} -\frac {27 b x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {b x^2}{2},-\frac {b x^2}{18}\right )}{d \sqrt [3]{b x^2+2} \left (b x^2+18\right ) \left (b x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-\frac {b x^2}{2},-\frac {b x^2}{18}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-\frac {b x^2}{2},-\frac {b x^2}{18}\right )\right )-27 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {b x^2}{2},-\frac {b x^2}{18}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 5.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{2+b x^2} \left (\frac {18 d}{b}+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 (b x^{2} + 2\right )}^{\frac {1}{3}} {\left (d x^{2} + \frac {18 \, d}{b}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{2}+2\right )^{\frac {1}{3}} \left (d \,x^{2}+\frac {18 d}{b}\right )}\, 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 (b x^{2} + 2\right )}^{\frac {1}{3}} {\left (d x^{2} + \frac {18 \, d}{b}\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 (\frac {18\,d}{b}+d\,x^2\right )\,{\left (b\,x^2+2\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*} \frac {b \int \frac {1}{b x^{2} \sqrt [3]{b x^{2} + 2} + 18 \sqrt [3]{b x^{2} + 2}}\, dx}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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