Optimal. Leaf size=38 \[ -\frac{\log \left (b+c x^2\right )}{2 b^2}+\frac{\log (x)}{b^2}+\frac{1}{2 b \left (b+c x^2\right )} \]
[Out]
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Rubi [A] time = 0.0718247, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{\log \left (b+c x^2\right )}{2 b^2}+\frac{\log (x)}{b^2}+\frac{1}{2 b \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[x^3/(b*x^2 + c*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 11.9673, size = 34, normalized size = 0.89 \[ \frac{1}{2 b \left (b + c x^{2}\right )} + \frac{\log{\left (x^{2} \right )}}{2 b^{2}} - \frac{\log{\left (b + c x^{2} \right )}}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(c*x**4+b*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.0219342, size = 33, normalized size = 0.87 \[ \frac{\frac{b}{b+c x^2}-\log \left (b+c x^2\right )+2 \log (x)}{2 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(b*x^2 + c*x^4)^2,x]
[Out]
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Maple [A] time = 0.017, size = 35, normalized size = 0.9 \[{\frac{1}{2\,b \left ( c{x}^{2}+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{2}}}-{\frac{\ln \left ( c{x}^{2}+b \right ) }{2\,{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(c*x^4+b*x^2)^2,x)
[Out]
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Maxima [A] time = 0.702543, size = 50, normalized size = 1.32 \[ \frac{1}{2 \,{\left (b c x^{2} + b^{2}\right )}} - \frac{\log \left (c x^{2} + b\right )}{2 \, b^{2}} + \frac{\log \left (x^{2}\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(c*x^4 + b*x^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.260859, size = 63, normalized size = 1.66 \[ -\frac{{\left (c x^{2} + b\right )} \log \left (c x^{2} + b\right ) - 2 \,{\left (c x^{2} + b\right )} \log \left (x\right ) - b}{2 \,{\left (b^{2} c x^{2} + b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(c*x^4 + b*x^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.65951, size = 34, normalized size = 0.89 \[ \frac{1}{2 b^{2} + 2 b c x^{2}} + \frac{\log{\left (x \right )}}{b^{2}} - \frac{\log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(c*x**4+b*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.271521, size = 49, normalized size = 1.29 \[ -\frac{{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b^{2}} + \frac{1}{2 \,{\left (c x^{2} + b\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(c*x^4 + b*x^2)^2,x, algorithm="giac")
[Out]