Optimal. Leaf size=202 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a+b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}} \]
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Rubi [A] time = 0.03, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {392} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a+b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 392
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 {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 166, normalized size = 0.82 \[ \frac {9 a x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2} \left (2 b x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )-F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )\right )+9 a F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int -\frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (b x^{2} - 3 \, a\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (-b \,x^{2}+3 a \right ) \left (b \,x^{2}+a \right )^{\frac {1}{3}}}\, dx \]
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 \[ -\int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (b x^{2} - 3 \, a\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^2+a\right )}^{1/3}\,\left (3\,a-b\,x^2\right )} \,d x \]
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 \[ - \int \frac {1}{- 3 a \sqrt [3]{a + b x^{2}} + b x^{2} \sqrt [3]{a + b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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