3.43 \(\int \frac{1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx\)

Optimal. Leaf size=153 \[ \frac{2 \tanh ^{-1}\left (\frac{d+4 e x}{\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}}\right )}{\sqrt{d^4-64 a e^3} \sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}}-\frac{2 \tanh ^{-1}\left (\frac{d+4 e x}{\sqrt{2 \sqrt{d^4-64 a e^3}+3 d^2}}\right )}{\sqrt{d^4-64 a e^3} \sqrt{2 \sqrt{d^4-64 a e^3}+3 d^2}} \]

[Out]

(2*ArcTanh[(d + 4*e*x)/Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]])/(Sqrt[d^4 - 64*a*e^3]*Sqrt[3*d^2 - 2*Sqrt[d^4 -
64*a*e^3]]) - (2*ArcTanh[(d + 4*e*x)/Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]]])/(Sqrt[d^4 - 64*a*e^3]*Sqrt[3*d^2 +
2*Sqrt[d^4 - 64*a*e^3]])

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Rubi [A]  time = 0.254435, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {1106, 1093, 208} \[ \frac{2 \tanh ^{-1}\left (\frac{d+4 e x}{\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}}\right )}{\sqrt{d^4-64 a e^3} \sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}}-\frac{2 \tanh ^{-1}\left (\frac{d+4 e x}{\sqrt{2 \sqrt{d^4-64 a e^3}+3 d^2}}\right )}{\sqrt{d^4-64 a e^3} \sqrt{2 \sqrt{d^4-64 a e^3}+3 d^2}} \]

Antiderivative was successfully verified.

[In]

Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-1),x]

[Out]

(2*ArcTanh[(d + 4*e*x)/Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]])/(Sqrt[d^4 - 64*a*e^3]*Sqrt[3*d^2 - 2*Sqrt[d^4 -
64*a*e^3]]) - (2*ArcTanh[(d + 4*e*x)/Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]]])/(Sqrt[d^4 - 64*a*e^3]*Sqrt[3*d^2 +
2*Sqrt[d^4 - 64*a*e^3]])

Rule 1106

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 1093

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

Rule 208

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

Rubi steps

\begin{align*} \int \frac{1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\frac{1}{32} \left (\frac{5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4} \, dx,x,\frac{d}{4 e}+x\right )\\ &=\frac{\left (4 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3 d^2 e}{2}-e \sqrt{d^4-64 a e^3}+8 e^3 x^2} \, dx,x,\frac{d}{4 e}+x\right )}{\sqrt{d^4-64 a e^3}}-\frac{\left (4 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3 d^2 e}{2}+e \sqrt{d^4-64 a e^3}+8 e^3 x^2} \, dx,x,\frac{d}{4 e}+x\right )}{\sqrt{d^4-64 a e^3}}\\ &=\frac{2 \tanh ^{-1}\left (\frac{d+4 e x}{\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}}\right )}{\sqrt{d^4-64 a e^3} \sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}}-\frac{2 \tanh ^{-1}\left (\frac{d+4 e x}{\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}}\right )}{\sqrt{d^4-64 a e^3} \sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}}\\ \end{align*}

Mathematica [C]  time = 0.0224046, size = 71, normalized size = 0.46 \[ -\text{RootSum}\left [8 \text{$\#$1}^3 d e^2+8 \text{$\#$1}^4 e^3-\text{$\#$1} d^3+8 a e^2\& ,\frac{\log (x-\text{$\#$1})}{-24 \text{$\#$1}^2 d e^2-32 \text{$\#$1}^3 e^3+d^3}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-1),x]

[Out]

-RootSum[8*a*e^2 - d^3*#1 + 8*d*e^2*#1^3 + 8*e^3*#1^4 & , Log[x - #1]/(d^3 - 24*d*e^2*#1^2 - 32*e^3*#1^3) & ]

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Maple [C]  time = 0.067, size = 67, normalized size = 0.4 \begin{align*} \sum _{{\it \_R}={\it RootOf} \left ( 8\,{e}^{3}{{\it \_Z}}^{4}+8\,d{e}^{2}{{\it \_Z}}^{3}-{d}^{3}{\it \_Z}+8\,a{e}^{2} \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{32\,{{\it \_R}}^{3}{e}^{3}+24\,{{\it \_R}}^{2}d{e}^{2}-{d}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x)

[Out]

sum(1/(32*_R^3*e^3+24*_R^2*d*e^2-d^3)*ln(x-_R),_R=RootOf(8*_Z^4*e^3+8*_Z^3*d*e^2-_Z*d^3+8*a*e^2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x, algorithm="maxima")

[Out]

integrate(1/(8*e^3*x^4 + 8*d*e^2*x^3 - d^3*x + 8*a*e^2), x)

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Fricas [B]  time = 1.6947, size = 2630, normalized size = 17.19 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x, algorithm="fricas")

[Out]

-sqrt((3*d^2 + 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 419
4304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*x + 2*(2*d^4 - 128*a*e^3 - 3*(5*d^10 - 64*a*d^6
*e^3 - 16384*a^2*d^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))*sqrt((3*d^2 + 2
*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(
5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)) + 2*d) + sqrt((3*d^2 + 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25
*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*
x - 2*(2*d^4 - 128*a*e^3 - 3*(5*d^10 - 64*a*d^6*e^3 - 16384*a^2*d^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*
a^2*d^4*e^6 - 4194304*a^3*e^9))*sqrt((3*d^2 + 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^
8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)) + 2*d) - sqrt((3*d^2 - 2
*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(
5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*x + 2*(2*d^4 - 128*a*e^3 + 3*(5*d^10 - 64*a*d^6*e^3 - 16384*a^2
*d^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))*sqrt((3*d^2 - 2*(5*d^8 - 64*a*d
^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4
*e^3 - 16384*a^2*e^6)) + 2*d) + sqrt((3*d^2 - 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^
8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*x - 2*(2*d^4 - 1
28*a*e^3 + 3*(5*d^10 - 64*a*d^6*e^3 - 16384*a^2*d^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 41
94304*a^3*e^9))*sqrt((3*d^2 - 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^
2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)) + 2*d)

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Sympy [A]  time = 1.45378, size = 122, normalized size = 0.8 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (1048576 a^{3} e^{9} - 12288 a^{2} d^{4} e^{6} - 384 a d^{8} e^{3} + 5 d^{12}\right ) + t^{2} \left (384 a d^{2} e^{3} - 6 d^{6}\right ) + 1, \left ( t \mapsto t \log{\left (x + \frac{- 49152 t^{3} a^{2} d^{2} e^{6} - 192 t^{3} a d^{6} e^{3} + 15 t^{3} d^{10} + 256 t a e^{3} - 13 t d^{4} + 2 d}{8 e} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2),x)

[Out]

RootSum(_t**4*(1048576*a**3*e**9 - 12288*a**2*d**4*e**6 - 384*a*d**8*e**3 + 5*d**12) + _t**2*(384*a*d**2*e**3
- 6*d**6) + 1, Lambda(_t, _t*log(x + (-49152*_t**3*a**2*d**2*e**6 - 192*_t**3*a*d**6*e**3 + 15*_t**3*d**10 + 2
56*_t*a*e**3 - 13*_t*d**4 + 2*d)/(8*e))))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x, algorithm="giac")

[Out]

integrate(1/(8*e^3*x^4 + 8*d*e^2*x^3 - d^3*x + 8*a*e^2), x)