Optimal. Leaf size=147 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt [6]{2} \sqrt{3} \left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )}{\sqrt{b} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt{b} x}\right )}{12\ 2^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{2}}\right )}{12\ 2^{5/6} d} \]
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Rubi [A] time = 0.0797945, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt [6]{2} \sqrt{3} \left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )}{\sqrt{b} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt{b} x}\right )}{12\ 2^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{2}}\right )}{12\ 2^{5/6} d} \]
Antiderivative was successfully verified.
[In] Int[1/((-2 + b*x^2)^(1/3)*((-18*d)/b + d*x^2)),x]
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Rubi in Sympy [A] time = 25.286, size = 46, normalized size = 0.31 \[ \frac{b x \left (b x^{2} - 2\right )^{\frac{2}{3}} \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{3},1,\frac{3}{2},\frac{b x^{2}}{2},\frac{b x^{2}}{18} \right )}}{36 d \left (- \frac{b x^{2}}{2} + 1\right )^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2-2)**(1/3)/(-18*d/b+d*x**2),x)
[Out]
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Mathematica [C] time = 0.273102, size = 148, normalized size = 1.01 \[ \frac{27 b x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )}{d \left (b x^2-18\right ) \sqrt [3]{b x^2-2} \left (b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )\right )+27 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((-2 + b*x^2)^(1/3)*((-18*d)/b + d*x^2)),x]
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Maple [F] time = 0.063, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [3]{b{x}^{2}-2}}} \left ( -18\,{\frac{d}{b}}+d{x}^{2} \right ) ^{-1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2-2)^(1/3)/(-18/b*d+d*x^2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} - 2\right )}^{\frac{1}{3}}{\left (d x^{2} - \frac{18 \, d}{b}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 - 2)^(1/3)*(d*x^2 - 18*d/b)),x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 - 2)^(1/3)*(d*x^2 - 18*d/b)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b \int \frac{1}{b x^{2} \sqrt [3]{b x^{2} - 2} - 18 \sqrt [3]{b x^{2} - 2}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2-2)**(1/3)/(-18*d/b+d*x**2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} - 2\right )}^{\frac{1}{3}}{\left (d x^{2} - \frac{18 \, d}{b}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 - 2)^(1/3)*(d*x^2 - 18*d/b)),x, algorithm="giac")
[Out]