Optimal. Leaf size=40 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^2}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}} \]
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Rubi [A] time = 0.145437, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^2}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
[In] Int[(x*(2*e - 2*f*x^3))/(e^2 + 4*e*f*x^3 - 4*d*f*x^4 + 4*f^2*x^6),x]
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Rubi in Sympy [A] time = 50.0624, size = 41, normalized size = 1.02 \[ - \frac{\operatorname{atanh}{\left (\frac{4 \sqrt{d} \sqrt{f} x^{2}}{- 2 e - 4 f x^{3}} \right )}}{2 \sqrt{d} \sqrt{f}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(-2*f*x**3+2*e)/(4*f**2*x**6-4*d*f*x**4+4*e*f*x**3+e**2),x)
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Mathematica [C] time = 0.0740578, size = 86, normalized size = 2.15 \[ -\frac{\text{RootSum}\left [4 \text{$\#$1}^6 f^2-4 \text{$\#$1}^4 d f+4 \text{$\#$1}^3 e f+e^2\&,\frac{\text{$\#$1}^3 f \log (x-\text{$\#$1})-e \log (x-\text{$\#$1})}{6 \text{$\#$1}^4 f-4 \text{$\#$1}^2 d+3 \text{$\#$1} e}\&\right ]}{2 f} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(2*e - 2*f*x^3))/(e^2 + 4*e*f*x^3 - 4*d*f*x^4 + 4*f^2*x^6),x]
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Maple [C] time = 0.013, size = 74, normalized size = 1.9 \[ -{\frac{1}{2\,f}\sum _{{\it \_R}={\it RootOf} \left ( 4\,{f}^{2}{{\it \_Z}}^{6}-4\,df{{\it \_Z}}^{4}+4\,ef{{\it \_Z}}^{3}+{e}^{2} \right ) }{\frac{ \left ({{\it \_R}}^{4}f-{\it \_R}\,e \right ) \ln \left ( x-{\it \_R} \right ) }{6\,f{{\it \_R}}^{5}-4\,d{{\it \_R}}^{3}+3\,e{{\it \_R}}^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(-2*f*x^3+2*e)/(4*f^2*x^6-4*d*f*x^4+4*e*f*x^3+e^2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -2 \, \int \frac{{\left (f x^{3} - e\right )} x}{4 \, f^{2} x^{6} - 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-2*(f*x^3 - e)*x/(4*f^2*x^6 - 4*d*f*x^4 + 4*e*f*x^3 + e^2),x, algorithm="maxima")
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Fricas [A] time = 0.302629, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{8 \, d f^{2} x^{5} + 4 \, d e f x^{2} +{\left (4 \, f^{2} x^{6} + 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}\right )} \sqrt{d f}}{4 \, f^{2} x^{6} - 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}}\right )}{4 \, \sqrt{d f}}, \frac{\arctan \left (\frac{\sqrt{-d f} x}{d}\right ) + \arctan \left (\frac{2 \, f^{2} x^{4} - 2 \, d f x^{2} + e f x}{\sqrt{-d f} e}\right )}{2 \, \sqrt{-d f}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-2*(f*x^3 - e)*x/(4*f^2*x^6 - 4*d*f*x^4 + 4*e*f*x^3 + e^2),x, algorithm="fricas")
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Sympy [A] time = 4.44465, size = 66, normalized size = 1.65 \[ - \frac{\sqrt{\frac{1}{d f}} \log{\left (- d x^{2} \sqrt{\frac{1}{d f}} + \frac{e}{2 f} + x^{3} \right )}}{4} + \frac{\sqrt{\frac{1}{d f}} \log{\left (d x^{2} \sqrt{\frac{1}{d f}} + \frac{e}{2 f} + x^{3} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(-2*f*x**3+2*e)/(4*f**2*x**6-4*d*f*x**4+4*e*f*x**3+e**2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{2 \,{\left (f x^{3} - e\right )} x}{4 \, f^{2} x^{6} - 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-2*(f*x^3 - e)*x/(4*f^2*x^6 - 4*d*f*x^4 + 4*e*f*x^3 + e^2),x, algorithm="giac")
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