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 = 0.0338085, antiderivative size = 151, normalized size of antiderivative = 1., 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} \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\left (\sqrt [3]{2}-\sqrt [3]{b x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt{b} x}\right )}{12\ 2^{5/6} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt [6]{2} \sqrt{3} \left (\sqrt [3]{2}-\sqrt [3]{b x^2+2}\right )}{\sqrt{b} x}\right )}{4\ 2^{5/6} \sqrt{3} d}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{2}}\right )}{12\ 2^{5/6} d} \]
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*}
Mathematica [C] time = 0.145696, size = 148, normalized size = 0.98 \[ -\frac{27 b x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{2},-\frac{b x^2}{18}\right )}{d \sqrt [3]{b x^2+2} \left (b x^2+18\right ) \left (b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{18}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{18}\right )\right )-27 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{2},-\frac{b x^2}{18}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{b{x}^{2}+2}}} \left ( 18\,{\frac{d}{b}}+d{x}^{2} \right ) ^{-1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + 2\right )}^{\frac{1}{3}}{\left (d x^{2} + \frac{18 \, d}{b}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{b \int \frac{1}{b x^{2} \sqrt [3]{b x^{2} + 2} + 18 \sqrt [3]{b x^{2} + 2}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + 2\right )}^{\frac{1}{3}}{\left (d x^{2} + \frac{18 \, d}{b}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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