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 = 0.103011, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a+b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/((3*a - b*x^2)*(a + b*x^2)^(1/3)),x]
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Rubi in Sympy [A] time = 35.6769, size = 348, normalized size = 1.72 \[ \frac{\sqrt [3]{2} \sqrt{3} \log{\left (\sqrt{3} - \frac{\sqrt{b} x}{\sqrt{a}} \right )}}{24 a^{\frac{5}{6}} \sqrt{b}} - \frac{\sqrt [3]{2} \sqrt{3} \log{\left (\sqrt{3} + \frac{\sqrt{b} x}{\sqrt{a}} \right )}}{24 a^{\frac{5}{6}} \sqrt{b}} + \frac{\sqrt [3]{2} \sqrt{3} \log{\left (\sqrt{3} b - \frac{b^{\frac{3}{2}} x}{\sqrt{a}} - \frac{\sqrt [3]{2} \sqrt{3} b \sqrt [3]{a + b x^{2}}}{\sqrt [3]{a}} \right )}}{24 a^{\frac{5}{6}} \sqrt{b}} - \frac{\sqrt [3]{2} \sqrt{3} \log{\left (\sqrt{3} b + \frac{b^{\frac{3}{2}} x}{\sqrt{a}} - \frac{\sqrt [3]{2} \sqrt{3} b \sqrt [3]{a + b x^{2}}}{\sqrt [3]{a}} \right )}}{24 a^{\frac{5}{6}} \sqrt{b}} - \frac{\sqrt [3]{2} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \left (\sqrt{3} \sqrt{a} - \sqrt{b} x\right )}{3 \sqrt [6]{a} \sqrt [3]{a + b x^{2}}} \right )}}{12 a^{\frac{5}{6}} \sqrt{b}} + \frac{\sqrt [3]{2} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \left (\sqrt{3} \sqrt{a} + \sqrt{b} x\right )}{3 \sqrt [6]{a} \sqrt [3]{a + b x^{2}}} \right )}}{12 a^{\frac{5}{6}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-b*x**2+3*a)/(b*x**2+a)**(1/3),x)
[Out]
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Mathematica [C] time = 0.233952, 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.
[In] Integrate[1/((3*a - b*x^2)*(a + b*x^2)^(1/3)),x]
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Maple [F] time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{1}{-b{x}^{2}+3\,a}{\frac{1}{\sqrt [3]{b{x}^{2}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-b*x^2+3*a)/(b*x^2+a)^(1/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (b x^{2} - 3 \, a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b*x^2 + a)^(1/3)*(b*x^2 - 3*a)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b*x^2 + a)^(1/3)*(b*x^2 - 3*a)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{- 3 a \sqrt [3]{a + b x^{2}} + b x^{2} \sqrt [3]{a + b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-b*x**2+3*a)/(b*x**2+a)**(1/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (b x^{2} - 3 \, a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b*x^2 + a)^(1/3)*(b*x^2 - 3*a)),x, algorithm="giac")
[Out]