Optimal. Leaf size=68 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{7/2}}+\frac{5 a}{2 b^3 x}+\frac{1}{2 b x^3 \left (a x^2+b\right )}-\frac{5}{6 b^2 x^3} \]
[Out]
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Rubi [A] time = 0.082769, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{7/2}}+\frac{5 a}{2 b^3 x}+\frac{1}{2 b x^3 \left (a x^2+b\right )}-\frac{5}{6 b^2 x^3} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^2)^2*x^8),x]
[Out]
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Rubi in Sympy [A] time = 14.9566, size = 61, normalized size = 0.9 \[ \frac{5 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 b^{\frac{7}{2}}} + \frac{5 a}{2 b^{3} x} + \frac{1}{2 b x^{3} \left (a x^{2} + b\right )} - \frac{5}{6 b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**2/x**8,x)
[Out]
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Mathematica [A] time = 0.0730313, size = 67, normalized size = 0.99 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{7/2}}+\frac{a^2 x}{2 b^3 \left (a x^2+b\right )}+\frac{2 a}{b^3 x}-\frac{1}{3 b^2 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^2)^2*x^8),x]
[Out]
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Maple [A] time = 0.01, size = 59, normalized size = 0.9 \[ -{\frac{1}{3\,{b}^{2}{x}^{3}}}+2\,{\frac{a}{{b}^{3}x}}+{\frac{x{a}^{2}}{2\,{b}^{3} \left ( a{x}^{2}+b \right ) }}+{\frac{5\,{a}^{2}}{2\,{b}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^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)^2*x^8),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240285, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, a^{2} x^{4} + 20 \, a b x^{2} + 15 \,{\left (a^{2} x^{5} + a b x^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) - 4 \, b^{2}}{12 \,{\left (a b^{3} x^{5} + b^{4} x^{3}\right )}}, \frac{15 \, a^{2} x^{4} + 10 \, a b x^{2} + 15 \,{\left (a^{2} x^{5} + a b x^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right ) - 2 \, b^{2}}{6 \,{\left (a b^{3} x^{5} + b^{4} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^2*x^8),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.1533, size = 114, normalized size = 1.68 \[ - \frac{5 \sqrt{- \frac{a^{3}}{b^{7}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a^{3}}{b^{7}}}}{a^{2}} \right )}}{4} + \frac{5 \sqrt{- \frac{a^{3}}{b^{7}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a^{3}}{b^{7}}}}{a^{2}} \right )}}{4} + \frac{15 a^{2} x^{4} + 10 a b x^{2} - 2 b^{2}}{6 a b^{3} x^{5} + 6 b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**2/x**8,x)
[Out]
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GIAC/XCAS [A] time = 0.219144, size = 80, normalized size = 1.18 \[ \frac{5 \, a^{2} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{3}} + \frac{a^{2} x}{2 \,{\left (a x^{2} + b\right )} b^{3}} + \frac{6 \, a x^{2} - b}{3 \, b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^2*x^8),x, algorithm="giac")
[Out]