Optimal. Leaf size=49 \[ \frac{a \log \left (a x^2+b\right )}{b^3}-\frac{2 a \log (x)}{b^3}-\frac{a}{2 b^2 \left (a x^2+b\right )}-\frac{1}{2 b^2 x^2} \]
[Out]
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Rubi [A] time = 0.0978796, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a \log \left (a x^2+b\right )}{b^3}-\frac{2 a \log (x)}{b^3}-\frac{a}{2 b^2 \left (a x^2+b\right )}-\frac{1}{2 b^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^2)^2*x^7),x]
[Out]
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Rubi in Sympy [A] time = 12.1503, size = 46, normalized size = 0.94 \[ - \frac{a}{2 b^{2} \left (a x^{2} + b\right )} - \frac{a \log{\left (x^{2} \right )}}{b^{3}} + \frac{a \log{\left (a x^{2} + b \right )}}{b^{3}} - \frac{1}{2 b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**2/x**7,x)
[Out]
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Mathematica [A] time = 0.0633077, size = 41, normalized size = 0.84 \[ -\frac{b \left (\frac{a}{a x^2+b}+\frac{1}{x^2}\right )-2 a \log \left (a x^2+b\right )+4 a \log (x)}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^2)^2*x^7),x]
[Out]
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Maple [A] time = 0.019, size = 46, normalized size = 0.9 \[ -{\frac{1}{2\,{b}^{2}{x}^{2}}}-{\frac{a}{2\,{b}^{2} \left ( a{x}^{2}+b \right ) }}-2\,{\frac{a\ln \left ( x \right ) }{{b}^{3}}}+{\frac{a\ln \left ( a{x}^{2}+b \right ) }{{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^2/x^7,x)
[Out]
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Maxima [A] time = 1.44481, size = 70, normalized size = 1.43 \[ -\frac{2 \, a x^{2} + b}{2 \,{\left (a b^{2} x^{4} + b^{3} x^{2}\right )}} + \frac{a \log \left (a x^{2} + b\right )}{b^{3}} - \frac{a \log \left (x^{2}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^2*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224445, size = 99, normalized size = 2.02 \[ -\frac{2 \, a b x^{2} + b^{2} - 2 \,{\left (a^{2} x^{4} + a b x^{2}\right )} \log \left (a x^{2} + b\right ) + 4 \,{\left (a^{2} x^{4} + a b x^{2}\right )} \log \left (x\right )}{2 \,{\left (a b^{3} x^{4} + b^{4} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^2*x^7),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.07918, size = 49, normalized size = 1. \[ - \frac{2 a \log{\left (x \right )}}{b^{3}} + \frac{a \log{\left (x^{2} + \frac{b}{a} \right )}}{b^{3}} - \frac{2 a x^{2} + b}{2 a b^{2} x^{4} + 2 b^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**2/x**7,x)
[Out]
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GIAC/XCAS [A] time = 0.226984, size = 69, normalized size = 1.41 \[ -\frac{a{\rm ln}\left (x^{2}\right )}{b^{3}} + \frac{a{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{b^{3}} - \frac{2 \, a x^{2} + b}{2 \,{\left (a x^{4} + b x^{2}\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^2)^2*x^7),x, algorithm="giac")
[Out]