Optimal. Leaf size=40 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^3}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}} \]
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Rubi [A] time = 0.205808, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^3}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(3*e + 2*f*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 - 4*d*f*x^6),x]
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Rubi in Sympy [A] time = 68.4054, size = 37, normalized size = 0.92 \[ \frac{\operatorname{atanh}{\left (\frac{6 \sqrt{d} \sqrt{f} x^{3}}{3 e + 6 f x^{2}} \right )}}{2 \sqrt{d} \sqrt{f}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(2*f*x**2+3*e)/(-4*d*f*x**6+4*f**2*x**4+4*e*f*x**2+e**2),x)
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Mathematica [C] time = 0.0829876, size = 85, normalized size = 2.12 \[ \frac{\text{RootSum}\left [-4 \text{$\#$1}^6 d f+4 \text{$\#$1}^4 f^2+4 \text{$\#$1}^2 e f+e^2\&,\frac{2 \text{$\#$1}^3 f \log (x-\text{$\#$1})+3 \text{$\#$1} e \log (x-\text{$\#$1})}{-3 \text{$\#$1}^4 d+2 \text{$\#$1}^2 f+e}\&\right ]}{8 f} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(3*e + 2*f*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 - 4*d*f*x^6),x]
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Maple [C] time = 0.377, size = 77, normalized size = 1.9 \[ -{\frac{1}{8\,f}\sum _{{\it \_R}={\it RootOf} \left ( 4\,df{{\it \_Z}}^{6}-4\,{f}^{2}{{\it \_Z}}^{4}-4\,ef{{\it \_Z}}^{2}-{e}^{2} \right ) }{\frac{ \left ( 2\,{{\it \_R}}^{4}f+3\,{{\it \_R}}^{2}e \right ) \ln \left ( x-{\it \_R} \right ) }{3\,d{{\it \_R}}^{5}-2\,f{{\it \_R}}^{3}-e{\it \_R}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(2*f*x^2+3*e)/(-4*d*f*x^6+4*f^2*x^4+4*e*f*x^2+e^2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (2 \, f x^{2} + 3 \, e\right )} x^{2}}{4 \, d f x^{6} - 4 \, f^{2} x^{4} - 4 \, e f x^{2} - e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*f*x^2 + 3*e)*x^2/(4*d*f*x^6 - 4*f^2*x^4 - 4*e*f*x^2 - e^2),x, algorithm="maxima")
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Fricas [A] time = 0.288159, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{8 \, d f^{2} x^{5} + 4 \, d e f x^{3} +{\left (4 \, d f x^{6} + 4 \, f^{2} x^{4} + 4 \, e f x^{2} + e^{2}\right )} \sqrt{d f}}{4 \, d f x^{6} - 4 \, f^{2} x^{4} - 4 \, e f x^{2} - e^{2}}\right )}{4 \, \sqrt{d f}}, \frac{\arctan \left (\frac{\sqrt{-d f} x}{f}\right ) - \arctan \left (\frac{2 \, d f x^{3} -{\left (d e + 2 \, f^{2}\right )} x}{\sqrt{-d f} e}\right ) + \arctan \left (\frac{2 \,{\left (2 \, d f^{2} x^{5} - e f^{2} x -{\left (d e f + 2 \, f^{3}\right )} x^{3}\right )}}{\sqrt{-d f} e^{2}}\right )}{2 \, \sqrt{-d f}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*f*x^2 + 3*e)*x^2/(4*d*f*x^6 - 4*f^2*x^4 - 4*e*f*x^2 - e^2),x, algorithm="fricas")
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Sympy [A] time = 4.30452, size = 80, normalized size = 2. \[ - \frac{\sqrt{\frac{1}{d f}} \log{\left (- \frac{e \sqrt{\frac{1}{d f}}}{2} - f x^{2} \sqrt{\frac{1}{d f}} + x^{3} \right )}}{4} + \frac{\sqrt{\frac{1}{d f}} \log{\left (\frac{e \sqrt{\frac{1}{d f}}}{2} + f x^{2} \sqrt{\frac{1}{d f}} + x^{3} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(2*f*x**2+3*e)/(-4*d*f*x**6+4*f**2*x**4+4*e*f*x**2+e**2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (2 \, f x^{2} + 3 \, e\right )} x^{2}}{4 \, d f x^{6} - 4 \, f^{2} x^{4} - 4 \, e f x^{2} - e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*f*x^2 + 3*e)*x^2/(4*d*f*x^6 - 4*f^2*x^4 - 4*e*f*x^2 - e^2),x, algorithm="giac")
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