Integrand size = 30, antiderivative size = 153 \[ \int \frac {1}{\sqrt [3]{-a-b x^2} \left (\frac {9 a d}{b}+d x^2\right )} \, dx=-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d}-\frac {\sqrt {b} \arctan \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{-a-b x^2}\right )^2}{3 \sqrt [6]{a} \sqrt {b} x}\right )}{12 a^{5/6} d}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{-a-b x^2}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} a^{5/6} d} \]
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Time = 0.02 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {403} \[ \int \frac {1}{\sqrt [3]{-a-b x^2} \left (\frac {9 a d}{b}+d x^2\right )} \, dx=-\frac {\sqrt {b} \arctan \left (\frac {\left (\sqrt [3]{-a-b x^2}+\sqrt [3]{a}\right )^2}{3 \sqrt [6]{a} \sqrt {b} x}\right )}{12 a^{5/6} d}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{-a-b x^2}+\sqrt [3]{a}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} a^{5/6} d} \]
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Rule 403
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{-a-b x^2}\right )^2}{3 \sqrt [6]{a} \sqrt {b} x}\right )}{12 a^{5/6} d}+\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{-a-b x^2}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} a^{5/6} d} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 5.80 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt [3]{-a-b x^2} \left (\frac {9 a d}{b}+d x^2\right )} \, dx=\frac {27 a b x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {b x^2}{9 a}\right )}{d \sqrt [3]{-a-b x^2} \left (9 a+b x^2\right ) \left (27 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {b x^2}{9 a}\right )-2 b x^2 \left (\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {b x^2}{9 a}\right )+3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {b x^2}{9 a}\right )\right )\right )} \]
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\[\int \frac {1}{\left (-b \,x^{2}-a \right )^{\frac {1}{3}} \left (\frac {9 a d}{b}+d \,x^{2}\right )}d x\]
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Timed out. \[ \int \frac {1}{\sqrt [3]{-a-b x^2} \left (\frac {9 a d}{b}+d x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt [3]{-a-b x^2} \left (\frac {9 a d}{b}+d x^2\right )} \, dx=\frac {b \int \frac {1}{9 a \sqrt [3]{- a - b x^{2}} + b x^{2} \sqrt [3]{- a - b x^{2}}}\, dx}{d} \]
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\[ \int \frac {1}{\sqrt [3]{-a-b x^2} \left (\frac {9 a d}{b}+d x^2\right )} \, dx=\int { \frac {1}{{\left (-b x^{2} - a\right )}^{\frac {1}{3}} {\left (d x^{2} + \frac {9 \, a d}{b}\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{-a-b x^2} \left (\frac {9 a d}{b}+d x^2\right )} \, dx=\int { \frac {1}{{\left (-b x^{2} - a\right )}^{\frac {1}{3}} {\left (d x^{2} + \frac {9 \, a d}{b}\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{-a-b x^2} \left (\frac {9 a d}{b}+d x^2\right )} \, dx=\int \frac {1}{{\left (-b\,x^2-a\right )}^{1/3}\,\left (d\,x^2+\frac {9\,a\,d}{b}\right )} \,d x \]
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