Optimal. Leaf size=84 \[ \frac{2 b \log \left (a+b x^2\right )}{a^5}-\frac{4 b \log (x)}{a^5}-\frac{3 b}{2 a^4 \left (a+b x^2\right )}-\frac{1}{2 a^4 x^2}-\frac{b}{2 a^3 \left (a+b x^2\right )^2}-\frac{b}{6 a^2 \left (a+b x^2\right )^3} \]
[Out]
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Rubi [A] time = 0.185911, antiderivative size = 84, 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 \[ \frac{2 b \log \left (a+b x^2\right )}{a^5}-\frac{4 b \log (x)}{a^5}-\frac{3 b}{2 a^4 \left (a+b x^2\right )}-\frac{1}{2 a^4 x^2}-\frac{b}{2 a^3 \left (a+b x^2\right )^2}-\frac{b}{6 a^2 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]
[Out]
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Rubi in Sympy [A] time = 31.8662, size = 82, normalized size = 0.98 \[ - \frac{b}{6 a^{2} \left (a + b x^{2}\right )^{3}} - \frac{b}{2 a^{3} \left (a + b x^{2}\right )^{2}} - \frac{3 b}{2 a^{4} \left (a + b x^{2}\right )} - \frac{1}{2 a^{4} x^{2}} - \frac{2 b \log{\left (x^{2} \right )}}{a^{5}} + \frac{2 b \log{\left (a + b x^{2} \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.118648, size = 70, normalized size = 0.83 \[ -\frac{\frac{a \left (3 a^3+22 a^2 b x^2+30 a b^2 x^4+12 b^3 x^6\right )}{x^2 \left (a+b x^2\right )^3}-12 b \log \left (a+b x^2\right )+24 b \log (x)}{6 a^5} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]
[Out]
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Maple [A] time = 0.023, size = 77, normalized size = 0.9 \[ -{\frac{1}{2\,{a}^{4}{x}^{2}}}-{\frac{b}{6\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{b}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,b}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}-4\,{\frac{b\ln \left ( x \right ) }{{a}^{5}}}+2\,{\frac{b\ln \left ( b{x}^{2}+a \right ) }{{a}^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
[Out]
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Maxima [A] time = 0.687942, size = 134, normalized size = 1.6 \[ -\frac{12 \, b^{3} x^{6} + 30 \, a b^{2} x^{4} + 22 \, a^{2} b x^{2} + 3 \, a^{3}}{6 \,{\left (a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{6} + 3 \, a^{6} b x^{4} + a^{7} x^{2}\right )}} + \frac{2 \, b \log \left (b x^{2} + a\right )}{a^{5}} - \frac{2 \, b \log \left (x^{2}\right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^2*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269571, size = 220, normalized size = 2.62 \[ -\frac{12 \, a b^{3} x^{6} + 30 \, a^{2} b^{2} x^{4} + 22 \, a^{3} b x^{2} + 3 \, a^{4} - 12 \,{\left (b^{4} x^{8} + 3 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 24 \,{\left (b^{4} x^{8} + 3 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b^{3} x^{8} + 3 \, a^{6} b^{2} x^{6} + 3 \, a^{7} b x^{4} + a^{8} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^2*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.83549, size = 100, normalized size = 1.19 \[ - \frac{3 a^{3} + 22 a^{2} b x^{2} + 30 a b^{2} x^{4} + 12 b^{3} x^{6}}{6 a^{7} x^{2} + 18 a^{6} b x^{4} + 18 a^{5} b^{2} x^{6} + 6 a^{4} b^{3} x^{8}} - \frac{4 b \log{\left (x \right )}}{a^{5}} + \frac{2 b \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.271369, size = 126, normalized size = 1.5 \[ -\frac{2 \, b{\rm ln}\left (x^{2}\right )}{a^{5}} + \frac{2 \, b{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{5}} + \frac{4 \, b x^{2} - a}{2 \, a^{5} x^{2}} - \frac{22 \, b^{4} x^{6} + 75 \, a b^{3} x^{4} + 87 \, a^{2} b^{2} x^{2} + 35 \, a^{3} b}{6 \,{\left (b x^{2} + a\right )}^{3} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^2*x^3),x, algorithm="giac")
[Out]