3.377 \(\int \frac{x^2}{e^2+4 e f x^3+4 d f x^6+4 f^2 x^6} \, dx\)

Optimal. Leaf size=42 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^3 (d+f)+e\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}} \]

[Out]

ArcTan[(Sqrt[f]*(e + 2*(d + f)*x^3))/(Sqrt[d]*e)]/(6*Sqrt[d]*e*Sqrt[f])

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Rubi [A]  time = 0.124634, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^3 (d+f)+e\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(e^2 + 4*e*f*x^3 + 4*d*f*x^6 + 4*f^2*x^6),x]

[Out]

ArcTan[(Sqrt[f]*(e + 2*(d + f)*x^3))/(Sqrt[d]*e)]/(6*Sqrt[d]*e*Sqrt[f])

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Rubi in Sympy [A]  time = 47.9681, size = 37, normalized size = 0.88 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{f} \left (e + x^{3} \left (2 d + 2 f\right )\right )}{\sqrt{d} e} \right )}}{6 \sqrt{d} e \sqrt{f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(4*d*f*x**6+4*f**2*x**6+4*e*f*x**3+e**2),x)

[Out]

atan(sqrt(f)*(e + x**3*(2*d + 2*f))/(sqrt(d)*e))/(6*sqrt(d)*e*sqrt(f))

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Mathematica [A]  time = 0.0315529, size = 42, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^3 (d+f)+e\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(e^2 + 4*e*f*x^3 + 4*d*f*x^6 + 4*f^2*x^6),x]

[Out]

ArcTan[(Sqrt[f]*(e + 2*(d + f)*x^3))/(Sqrt[d]*e)]/(6*Sqrt[d]*e*Sqrt[f])

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Maple [A]  time = 0.003, size = 42, normalized size = 1. \[{\frac{1}{6\,e}\arctan \left ({\frac{2\, \left ( 4\,df+4\,{f}^{2} \right ){x}^{3}+4\,ef}{4\,e}{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(4*d*f*x^6+4*f^2*x^6+4*e*f*x^3+e^2),x)

[Out]

1/6/e/(d*f)^(1/2)*arctan(1/4*(2*(4*d*f+4*f^2)*x^3+4*e*f)/e/(d*f)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(4*d*f*x^6 + 4*f^2*x^6 + 4*e*f*x^3 + e^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312027, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{2 \, d e^{2} f + 4 \,{\left (d^{2} e f + d e f^{2}\right )} x^{3} +{\left (4 \,{\left (d^{2} f + 2 \, d f^{2} + f^{3}\right )} x^{6} + 4 \,{\left (d e f + e f^{2}\right )} x^{3} - d e^{2} + e^{2} f\right )} \sqrt{-d f}}{4 \,{\left (d f + f^{2}\right )} x^{6} + 4 \, e f x^{3} + e^{2}}\right )}{12 \, \sqrt{-d f} e}, \frac{\arctan \left (\frac{{\left (2 \,{\left (d + f\right )} x^{3} + e\right )} \sqrt{d f}}{d e}\right )}{6 \, \sqrt{d f} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(4*d*f*x^6 + 4*f^2*x^6 + 4*e*f*x^3 + e^2),x, algorithm="fricas")

[Out]

[1/12*log((2*d*e^2*f + 4*(d^2*e*f + d*e*f^2)*x^3 + (4*(d^2*f + 2*d*f^2 + f^3)*x^
6 + 4*(d*e*f + e*f^2)*x^3 - d*e^2 + e^2*f)*sqrt(-d*f))/(4*(d*f + f^2)*x^6 + 4*e*
f*x^3 + e^2))/(sqrt(-d*f)*e), 1/6*arctan((2*(d + f)*x^3 + e)*sqrt(d*f)/(d*e))/(s
qrt(d*f)*e)]

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Sympy [A]  time = 2.46519, size = 78, normalized size = 1.86 \[ \frac{- \frac{\sqrt{- \frac{1}{d f}} \log{\left (x^{3} + \frac{- d e \sqrt{- \frac{1}{d f}} + e}{2 d + 2 f} \right )}}{12} + \frac{\sqrt{- \frac{1}{d f}} \log{\left (x^{3} + \frac{d e \sqrt{- \frac{1}{d f}} + e}{2 d + 2 f} \right )}}{12}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(4*d*f*x**6+4*f**2*x**6+4*e*f*x**3+e**2),x)

[Out]

(-sqrt(-1/(d*f))*log(x**3 + (-d*e*sqrt(-1/(d*f)) + e)/(2*d + 2*f))/12 + sqrt(-1/
(d*f))*log(x**3 + (d*e*sqrt(-1/(d*f)) + e)/(2*d + 2*f))/12)/e

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GIAC/XCAS [A]  time = 0.27136, size = 51, normalized size = 1.21 \[ \frac{\arctan \left (\frac{{\left (2 \, d f x^{3} + 2 \, f^{2} x^{3} + f e\right )} e^{\left (-1\right )}}{\sqrt{d f}}\right ) e^{\left (-1\right )}}{6 \, \sqrt{d f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(4*d*f*x^6 + 4*f^2*x^6 + 4*e*f*x^3 + e^2),x, algorithm="giac")

[Out]

1/6*arctan((2*d*f*x^3 + 2*f^2*x^3 + f*e)*e^(-1)/sqrt(d*f))*e^(-1)/sqrt(d*f)