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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.142368, 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 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^2}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}} \]

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]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 48.5976, size = 41, normalized size = 1.02 \[ - \frac{\operatorname{atan}{\left (\frac{4 \sqrt{d} \sqrt{f} x^{2}}{- 2 e - 4 f x^{3}} \right )}}{2 \sqrt{d} \sqrt{f}} \]

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)

[Out]

-atan(4*sqrt(d)*sqrt(f)*x**2/(-2*e - 4*f*x**3))/(2*sqrt(d)*sqrt(f))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0768785, 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]

[Out]

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

_______________________________________________________________________________________

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)

[Out]

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

_______________________________________________________________________________________

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")

[Out]

-2*integrate((f*x^3 - e)*x/(4*f^2*x^6 + 4*d*f*x^4 + 4*e*f*x^3 + e^2), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.282742, 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")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 4.45304, size = 73, normalized size = 1.82 \[ \frac{\sqrt{- \frac{1}{d f}} \log{\left (- d x^{2} \sqrt{- \frac{1}{d f}} + \frac{e}{2 f} + x^{3} \right )}}{4} - \frac{\sqrt{- \frac{1}{d f}} \log{\left (d x^{2} \sqrt{- \frac{1}{d f}} + \frac{e}{2 f} + x^{3} \right )}}{4} \]

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]

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

_______________________________________________________________________________________

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")

[Out]

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)