Optimal. Leaf size=153 \[ -\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} a^{5/6} d}+\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}{3 \sqrt [6]{a} \sqrt {b} x}\right )}{12 a^{5/6} d}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d} \]
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Rubi [A] time = 0.03, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {395} \[ -\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} a^{5/6} d}+\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}{3 \sqrt [6]{a} \sqrt {b} x}\right )}{12 a^{5/6} d}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d} \]
Antiderivative was successfully verified.
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Rule 395
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{a-b x^2} \left (-\frac {9 a d}{b}+d x^2\right )} \, dx &=-\frac {\sqrt {b} \tan ^{-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}-\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{3 \sqrt {a}}\right )}{12 a^{5/6} d}+\frac {\sqrt {b} \tanh ^{-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}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 167, normalized size = 1.09 \[ -\frac {27 a b x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {b x^2}{a},\frac {b x^2}{9 a}\right )}{d \sqrt [3]{a-b x^2} \left (9 a-b x^2\right ) \left (2 b x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};\frac {b x^2}{a},\frac {b x^2}{9 a}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};\frac {b x^2}{a},\frac {b x^2}{9 a}\right )\right )+27 a F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {b x^2}{a},\frac {b x^2}{9 a}\right )\right )} \]
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 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {1}{3}} {\left (d x^{2} - \frac {9 \, a d}{b}\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}} \left (d \,x^{2}-\frac {9 a d}{b}\right )}\, dx \]
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 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {1}{3}} {\left (d x^{2} - \frac {9 \, a d}{b}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a-b\,x^2\right )}^{1/3}\,\left (d\,x^2-\frac {9\,a\,d}{b}\right )} \,d x \]
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 \[ \frac {b \int \frac {1}{- 9 a \sqrt [3]{a - b x^{2}} + b x^{2} \sqrt [3]{a - b x^{2}}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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