Optimal. Leaf size=58 \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}-\frac{a^2}{b^3 x}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]
[Out]
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Rubi [A] time = 0.0835069, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}-\frac{a^2}{b^3 x}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^2)*x^8),x]
[Out]
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Rubi in Sympy [A] time = 15.4592, size = 49, normalized size = 0.84 \[ - \frac{a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{b^{\frac{7}{2}}} - \frac{a^{2}}{b^{3} x} + \frac{a}{3 b^{2} x^{3}} - \frac{1}{5 b x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)/x**8,x)
[Out]
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Mathematica [A] time = 0.0500393, size = 58, normalized size = 1. \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}-\frac{a^2}{b^3 x}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^2)*x^8),x]
[Out]
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Maple [A] time = 0.006, size = 52, normalized size = 0.9 \[ -{\frac{1}{5\,b{x}^{5}}}-{\frac{{a}^{2}}{{b}^{3}x}}+{\frac{a}{3\,{b}^{2}{x}^{3}}}-{\frac{{a}^{3}}{{b}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)/x^8,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)*x^8),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229053, size = 1, normalized size = 0.02 \[ \left [\frac{15 \, a^{2} x^{5} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) - 30 \, a^{2} x^{4} + 10 \, a b x^{2} - 6 \, b^{2}}{30 \, b^{3} x^{5}}, -\frac{15 \, a^{2} x^{5} \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right ) + 15 \, a^{2} x^{4} - 5 \, a b x^{2} + 3 \, b^{2}}{15 \, b^{3} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)*x^8),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.79658, size = 100, normalized size = 1.72 \[ \frac{\sqrt{- \frac{a^{5}}{b^{7}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{7}}}}{a^{3}} \right )}}{2} - \frac{\sqrt{- \frac{a^{5}}{b^{7}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{7}}}}{a^{3}} \right )}}{2} - \frac{15 a^{2} x^{4} - 5 a b x^{2} + 3 b^{2}}{15 b^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)/x**8,x)
[Out]
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GIAC/XCAS [A] time = 0.225045, size = 70, normalized size = 1.21 \[ -\frac{a^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{15 \, a^{2} x^{4} - 5 \, a b x^{2} + 3 \, b^{2}}{15 \, b^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)*x^8),x, algorithm="giac")
[Out]