Optimal. Leaf size=57 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{5/2}}+\frac{1}{2 b x \left (a x^2+b\right )}-\frac{3}{2 b^2 x} \]
[Out]
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Rubi [A] time = 0.0660439, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{5/2}}+\frac{1}{2 b x \left (a x^2+b\right )}-\frac{3}{2 b^2 x} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^2)^2*x^6),x]
[Out]
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Rubi in Sympy [A] time = 10.9878, size = 48, normalized size = 0.84 \[ - \frac{3 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 b^{\frac{5}{2}}} + \frac{1}{2 b x \left (a x^{2} + b\right )} - \frac{3}{2 b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**2/x**6,x)
[Out]
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Mathematica [A] time = 0.0651415, size = 54, normalized size = 0.95 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{5/2}}-\frac{a x}{2 b^2 \left (a x^2+b\right )}-\frac{1}{b^2 x} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^2)^2*x^6),x]
[Out]
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Maple [A] time = 0.01, size = 46, normalized size = 0.8 \[ -{\frac{1}{{b}^{2}x}}-{\frac{ax}{2\,{b}^{2} \left ( a{x}^{2}+b \right ) }}-{\frac{3\,a}{2\,{b}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^2/x^6,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^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233679, size = 1, normalized size = 0.02 \[ \left [-\frac{6 \, a x^{2} - 3 \,{\left (a x^{3} + b x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) + 4 \, b}{4 \,{\left (a b^{2} x^{3} + b^{3} x\right )}}, -\frac{3 \, a x^{2} + 3 \,{\left (a x^{3} + b x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right ) + 2 \, b}{2 \,{\left (a b^{2} x^{3} + b^{3} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^2*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.75089, size = 90, normalized size = 1.58 \[ \frac{3 \sqrt{- \frac{a}{b^{5}}} \log{\left (x - \frac{b^{3} \sqrt{- \frac{a}{b^{5}}}}{a} \right )}}{4} - \frac{3 \sqrt{- \frac{a}{b^{5}}} \log{\left (x + \frac{b^{3} \sqrt{- \frac{a}{b^{5}}}}{a} \right )}}{4} - \frac{3 a x^{2} + 2 b}{2 a b^{2} x^{3} + 2 b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**2/x**6,x)
[Out]
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GIAC/XCAS [A] time = 0.224528, size = 63, normalized size = 1.11 \[ -\frac{3 \, a \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} - \frac{3 \, a x^{2} + 2 \, b}{2 \,{\left (a x^{3} + b x\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^2*x^6),x, algorithm="giac")
[Out]