Integrand size = 24, antiderivative size = 202 \[ \int \frac {1}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}} \]
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Time = 0.02 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {401} \[ \int \frac {1}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a+b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}} \]
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Rule 401
Rubi steps \begin{align*} \text {integral}& = -\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*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 5.14 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2}} \, dx=\frac {9 a x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2} \left (9 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )+2 b x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )-\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-\frac {b x^2}{a},\frac {b x^2}{3 a}\right )\right )\right )} \]
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\[\int \frac {1}{\left (-b \,x^{2}+3 a \right ) \left (b \,x^{2}+a \right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {1}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2}} \, dx=- \int \frac {1}{- 3 a \sqrt [3]{a + b x^{2}} + b x^{2} \sqrt [3]{a + b x^{2}}}\, dx \]
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\[ \int \frac {1}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2}} \, dx=\int { -\frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (b x^{2} - 3 \, a\right )}} \,d x } \]
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\[ \int \frac {1}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2}} \, dx=\int { -\frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{3}} {\left (b x^{2} - 3 \, a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{1/3}\,\left (3\,a-b\,x^2\right )} \,d x \]
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