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

Optimal. Leaf size=44 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \left (e-2 x^3 (d-f)\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}} \]

[Out]

-ArcTanh[(Sqrt[f]*(e - 2*(d - f)*x^3))/(Sqrt[d]*e)]/(6*Sqrt[d]*e*Sqrt[f])

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Rubi [A]  time = 0.0616221, antiderivative size = 44, 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, Rules used = {6, 1352, 618, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \left (e-2 x^3 (d-f)\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}} \]

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]

-ArcTanh[(Sqrt[f]*(e - 2*(d - f)*x^3))/(Sqrt[d]*e)]/(6*Sqrt[d]*e*Sqrt[f])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 1352

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*x +
c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^2}{e^2+4 e f x^3-4 d f x^6+4 f^2 x^6} \, dx &=\int \frac{x^2}{e^2+4 e f x^3+\left (-4 d f+4 f^2\right ) x^6} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{e^2+4 e f x+\left (-4 d f+4 f^2\right ) x^2} \, dx,x,x^3\right )\\ &=-\left (\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{16 d e^2 f-x^2} \, dx,x,4 f \left (e-2 (d-f) x^3\right )\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \left (e-2 (d-f) x^3\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}}\\ \end{align*}

Mathematica [A]  time = 0.0218638, size = 46, normalized size = 1.05 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \left (-2 d x^3+e+2 f x^3\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}} \]

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]

-ArcTanh[(Sqrt[f]*(e - 2*d*x^3 + 2*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. \begin{align*}{\frac{1}{6\,e}{\it Artanh} \left ({\frac{2\, \left ( 4\,df-4\,{f}^{2} \right ){x}^{3}-4\,fe}{4\,e}{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}} \end{align*}

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)*arctanh(1/4*(2*(4*d*f-4*f^2)*x^3-4*f*e)/e/(d*f)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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 = 1.28025, size = 343, normalized size = 7.8 \begin{align*} \left [\frac{\sqrt{d f} \log \left (-\frac{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 + 2 \,{\left (2 \,{\left (d e - e f\right )} x^{3} - e^{2}\right )} \sqrt{d f}}{4 \,{\left (d f - f^{2}\right )} x^{6} - 4 \, e f x^{3} - e^{2}}\right )}{12 \, d e f}, \frac{\sqrt{-d f} \arctan \left (-\frac{{\left (2 \,{\left (d - f\right )} x^{3} - e\right )} \sqrt{-d f}}{d e}\right )}{6 \, d e f}\right ] \end{align*}

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

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Sympy [A]  time = 0.736859, size = 75, normalized size = 1.7 \begin{align*} - \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} \end{align*}

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 [A]  time = 1.75315, size = 55, normalized size = 1.25 \begin{align*} -\frac{\arctan \left (\frac{2 \, d f x^{3} - 2 \, f^{2} x^{3} - f e}{\sqrt{-d f e^{2}}}\right )}{6 \, \sqrt{-d f e^{2}}} \end{align*}

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)/sqrt(-d*f*e^2))/sqrt(-d*f*e^2)