Optimal. Leaf size=104 \[ \frac{\tan ^{-1}\left (\frac{\left (1-\sqrt [3]{b x^2+1}\right )^2}{3 \sqrt{b} x}\right )}{12 \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{b x^2+1}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{12 \sqrt{b}} \]
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Rubi [A] time = 0.0163418, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {394} \[ \frac{\tan ^{-1}\left (\frac{\left (1-\sqrt [3]{b x^2+1}\right )^2}{3 \sqrt{b} x}\right )}{12 \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{b x^2+1}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{12 \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 394
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{12 \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\left (1-\sqrt [3]{1+b x^2}\right )^2}{3 \sqrt{b} x}\right )}{12 \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{1+b x^2}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} \sqrt{b}}\\ \end{align*}
Mathematica [C] time = 0.108447, size = 137, normalized size = 1.32 \[ -\frac{27 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-b x^2,-\frac{b x^2}{9}\right )}{\sqrt [3]{b x^2+1} \left (b x^2+9\right ) \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-b x^2,-\frac{b x^2}{9}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-b x^2,-\frac{b x^2}{9}\right )\right )-27 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-b x^2,-\frac{b x^2}{9}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{2}+9}{\frac{1}{\sqrt [3]{b{x}^{2}+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} + 9\right )}{\left (b x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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*} \int \frac{1}{\sqrt [3]{b x^{2} + 1} \left (b x^{2} + 9\right )}\, dx \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} + 9\right )}{\left (b x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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