Optimal. Leaf size=66 \[ -\frac{3 b^2 \log \left (a+b x^2\right )}{2 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{b^2}{2 a^3 \left (a+b x^2\right )}+\frac{b}{a^3 x^2}-\frac{1}{4 a^2 x^4} \]
[Out]
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Rubi [A] time = 0.120071, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{3 b^2 \log \left (a+b x^2\right )}{2 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{b^2}{2 a^3 \left (a+b x^2\right )}+\frac{b}{a^3 x^2}-\frac{1}{4 a^2 x^4} \]
Antiderivative was successfully verified.
[In] Int[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 22.2745, size = 66, normalized size = 1. \[ - \frac{1}{4 a^{2} x^{4}} + \frac{b^{2}}{2 a^{3} \left (a + b x^{2}\right )} + \frac{b}{a^{3} x^{2}} + \frac{3 b^{2} \log{\left (x^{2} \right )}}{2 a^{4}} - \frac{3 b^{2} \log{\left (a + b x^{2} \right )}}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**5/(b**2*x**4+2*a*b*x**2+a**2),x)
[Out]
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Mathematica [A] time = 0.117276, size = 57, normalized size = 0.86 \[ \frac{-6 b^2 \log \left (a+b x^2\right )+a \left (\frac{2 b^2}{a+b x^2}-\frac{a}{x^4}+\frac{4 b}{x^2}\right )+12 b^2 \log (x)}{4 a^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]
[Out]
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Maple [A] time = 0.02, size = 61, normalized size = 0.9 \[ -{\frac{1}{4\,{a}^{2}{x}^{4}}}+{\frac{b}{{a}^{3}{x}^{2}}}+{\frac{{b}^{2}}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^5/(b^2*x^4+2*a*b*x^2+a^2),x)
[Out]
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Maxima [A] time = 0.689011, size = 95, normalized size = 1.44 \[ \frac{6 \, b^{2} x^{4} + 3 \, a b x^{2} - a^{2}}{4 \,{\left (a^{3} b x^{6} + a^{4} x^{4}\right )}} - \frac{3 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac{3 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.259477, size = 122, normalized size = 1.85 \[ \frac{6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3} - 6 \,{\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 12 \,{\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{4} b x^{6} + a^{5} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.42986, size = 68, normalized size = 1.03 \[ \frac{- a^{2} + 3 a b x^{2} + 6 b^{2} x^{4}}{4 a^{4} x^{4} + 4 a^{3} b x^{6}} + \frac{3 b^{2} \log{\left (x \right )}}{a^{4}} - \frac{3 b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**5/(b**2*x**4+2*a*b*x**2+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.269886, size = 116, normalized size = 1.76 \[ \frac{3 \, b^{2}{\rm ln}\left (x^{2}\right )}{2 \, a^{4}} - \frac{3 \, b^{2}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac{3 \, b^{3} x^{2} + 4 \, a b^{2}}{2 \,{\left (b x^{2} + a\right )} a^{4}} - \frac{9 \, b^{2} x^{4} - 4 \, a b x^{2} + a^{2}}{4 \, a^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^5),x, algorithm="giac")
[Out]