Optimal. Leaf size=252 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a} \left (\sqrt [3]{-a}-\sqrt [3]{2} \sqrt [3]{b x^2-a}\right )}{\sqrt [3]{-a} \sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{-a} \sqrt{b} x}{\sqrt{a} \left (\sqrt [3]{2} \sqrt [3]{b x^2-a}+\sqrt [3]{-a}\right )}\right )}{2\ 2^{2/3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{6\ 2^{2/3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}} \]
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Rubi [A] time = 0.0727463, antiderivative size = 252, 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 = {393} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a} \left (\sqrt [3]{-a}-\sqrt [3]{2} \sqrt [3]{b x^2-a}\right )}{\sqrt [3]{-a} \sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{-a} \sqrt{b} x}{\sqrt{a} \left (\sqrt [3]{2} \sqrt [3]{b x^2-a}+\sqrt [3]{-a}\right )}\right )}{2\ 2^{2/3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{6\ 2^{2/3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 393
Rubi steps
\begin{align*} \int \frac{1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a} \left (\sqrt [3]{-a}-\sqrt [3]{2} \sqrt [3]{-a+b x^2}\right )}{\sqrt [3]{-a} \sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{6\ 2^{2/3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{-a} \sqrt{b} x}{\sqrt{a} \left (\sqrt [3]{-a}+\sqrt [3]{2} \sqrt [3]{-a+b x^2}\right )}\right )}{2\ 2^{2/3} \sqrt [3]{-a} \sqrt{a} \sqrt{b}}\\ \end{align*}
Mathematica [C] time = 0.124108, size = 163, normalized size = 0.65 \[ -\frac{9 a x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{\sqrt [3]{b x^2-a} \left (3 a+b x^2\right ) \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-b{x}^{2}-3\,a}{\frac{1}{\sqrt [3]{b{x}^{2}-a}}}}\, 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} + 3 \, a\right )}{\left (b x^{2} - a\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}{3 a \sqrt [3]{- a + b x^{2}} + b x^{2} \sqrt [3]{- a + b x^{2}}}\, 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} + 3 \, a\right )}{\left (b x^{2} - a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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