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

Optimal. Leaf size=40 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^3}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}} \]

[Out]

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

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Rubi [A]  time = 0.204955, 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 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^3}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}} \]

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]

[Out]

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

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Rubi in Sympy [A]  time = 64.986, size = 37, normalized size = 0.92 \[ \frac{\operatorname{atan}{\left (\frac{6 \sqrt{d} \sqrt{f} x^{3}}{3 e + 6 f x^{2}} \right )}}{2 \sqrt{d} \sqrt{f}} \]

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)

[Out]

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

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Mathematica [C]  time = 0.0755973, 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]

[Out]

RootSum[e^2 + 4*e*f*#1^2 + 4*f^2*#1^4 + 4*d*f*#1^6 & , (3*e*Log[x - #1]*#1 + 2*f
*Log[x - #1]*#1^3)/(e + 2*f*#1^2 + 3*d*#1^4) & ]/(8*f)

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Maple [C]  time = 0.419, size = 74, 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)

[Out]

1/8/f*sum((2*_R^4*f+3*_R^2*e)/(3*_R^5*d+2*_R^3*f+_R*e)*ln(x-_R),_R=RootOf(4*_Z^6
*d*f+4*_Z^4*f^2+4*_Z^2*e*f+e^2))

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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")

[Out]

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)

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Fricas [A]  time = 0.286408, 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")

[Out]

[1/4*log((8*d*f^2*x^5 + 4*d*e*f*x^3 + (4*d*f*x^6 - 4*f^2*x^4 - 4*e*f*x^2 - e^2)*
sqrt(-d*f))/(4*d*f*x^6 + 4*f^2*x^4 + 4*e*f*x^2 + e^2))/sqrt(-d*f), 1/2*(arctan(s
qrt(d*f)*x/f) + arctan((2*d*f*x^3 - (d*e - 2*f^2)*x)/(sqrt(d*f)*e)) - arctan(2*(
2*d*f^2*x^5 + e*f^2*x - (d*e*f - 2*f^3)*x^3)/(sqrt(d*f)*e^2)))/sqrt(d*f)]

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Sympy [A]  time = 4.23546, size = 90, normalized size = 2.25 \[ - \frac{\sqrt{- \frac{1}{d f}} \log{\left (- \frac{e \sqrt{- \frac{1}{d f}}}{2} - f x^{2} \sqrt{- \frac{1}{d f}} + x^{3} \right )}}{4} + \frac{\sqrt{- \frac{1}{d f}} \log{\left (\frac{e \sqrt{- \frac{1}{d f}}}{2} + f x^{2} \sqrt{- \frac{1}{d f}} + x^{3} \right )}}{4} \]

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)

[Out]

-sqrt(-1/(d*f))*log(-e*sqrt(-1/(d*f))/2 - f*x**2*sqrt(-1/(d*f)) + x**3)/4 + sqrt
(-1/(d*f))*log(e*sqrt(-1/(d*f))/2 + f*x**2*sqrt(-1/(d*f)) + x**3)/4

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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")

[Out]

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)