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 = 0.07, antiderivative size = 252, 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 = {393} \begin {gather*} -\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}} \end {gather*}
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*}
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Mathematica [C] time = 0.14, size = 163, normalized size = 0.65 \begin {gather*} -\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 )} \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}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, 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} + 3 \, a\right )} {\left (b x^{2} - a\right )}^{\frac {1}{3}}}\,{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}-3 a \right ) \left (b \,x^{2}-a \right )^{\frac {1}{3}}}\, 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} + 3 \, a\right )} {\left (b x^{2} - a\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 = -0.00 \begin {gather*} -\int \frac {1}{{\left (b\,x^2-a\right )}^{1/3}\,\left (b\,x^2+3\,a\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*} - \int \frac {1}{3 a \sqrt [3]{- a + b x^{2}} + b x^{2} \sqrt [3]{- a + b x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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