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

Optimal. Leaf size=235 \[ \frac{\sqrt [4]{256 a e^3+5 d^4} \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{\sqrt{2} e \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \]

[Out]

((5*d^4 + 256*a*e^3)^(1/4)*Sqrt[(e*(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))/
((5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/Sqrt[5*d^4 + 256*a*e^3])^2)]*
(1 + (16*e^2*(d/(4*e) + x)^2)/Sqrt[5*d^4 + 256*a*e^3])*EllipticF[2*ArcTan[(d + 4
*e*x)/(5*d^4 + 256*a*e^3)^(1/4)], (1 + (3*d^2)/Sqrt[5*d^4 + 256*a*e^3])/2])/(Sqr
t[2]*e*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4])

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Rubi [A]  time = 0.402613, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{\sqrt [4]{256 a e^3+5 d^4} \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{\sqrt{2} e \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4],x]

[Out]

((5*d^4 + 256*a*e^3)^(1/4)*Sqrt[(e*(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))/
((5*d^4 + 256*a*e^3)*(1 + (d + 4*e*x)^2/Sqrt[5*d^4 + 256*a*e^3])^2)]*(1 + (d + 4
*e*x)^2/Sqrt[5*d^4 + 256*a*e^3])*EllipticF[2*ArcTan[(d + 4*e*x)/(5*d^4 + 256*a*e
^3)^(1/4)], (1 + (3*d^2)/Sqrt[5*d^4 + 256*a*e^3])/2])/(Sqrt[2]*e*Sqrt[8*a*e^2 -
d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4])

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Rubi in Sympy [A]  time = 60.406, size = 243, normalized size = 1.03 \[ \frac{\sqrt{2} \sqrt{\frac{256 a e^{3} + 5 d^{4} - 96 d^{2} e^{2} \left (\frac{d}{4 e} + x\right )^{2} + 256 e^{4} \left (\frac{d}{4 e} + x\right )^{4}}{\left (256 a e^{3} + 5 d^{4}\right ) \left (\frac{16 e^{2} \left (\frac{d}{4 e} + x\right )^{2}}{\sqrt{256 a e^{3} + 5 d^{4}}} + 1\right )^{2}}} \sqrt [4]{256 a e^{3} + 5 d^{4}} \left (\frac{16 e^{2} \left (\frac{d}{4 e} + x\right )^{2}}{\sqrt{256 a e^{3} + 5 d^{4}}} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{4 e \left (\frac{d}{4 e} + x\right )}{\sqrt [4]{256 a e^{3} + 5 d^{4}}} \right )}\middle | \frac{3 d^{2}}{2 \sqrt{256 a e^{3} + 5 d^{4}}} + \frac{1}{2}\right )}{2 e \sqrt{- 96 d^{2} e \left (\frac{d}{4 e} + x\right )^{2} + 256 e^{3} \left (\frac{d}{4 e} + x\right )^{4} + \frac{256 a e^{3} + 5 d^{4}}{e}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2)**(1/2),x)

[Out]

sqrt(2)*sqrt((256*a*e**3 + 5*d**4 - 96*d**2*e**2*(d/(4*e) + x)**2 + 256*e**4*(d/
(4*e) + x)**4)/((256*a*e**3 + 5*d**4)*(16*e**2*(d/(4*e) + x)**2/sqrt(256*a*e**3
+ 5*d**4) + 1)**2))*(256*a*e**3 + 5*d**4)**(1/4)*(16*e**2*(d/(4*e) + x)**2/sqrt(
256*a*e**3 + 5*d**4) + 1)*elliptic_f(2*atan(4*e*(d/(4*e) + x)/(256*a*e**3 + 5*d*
*4)**(1/4)), 3*d**2/(2*sqrt(256*a*e**3 + 5*d**4)) + 1/2)/(2*e*sqrt(-96*d**2*e*(d
/(4*e) + x)**2 + 256*e**3*(d/(4*e) + x)**4 + (256*a*e**3 + 5*d**4)/e))

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Mathematica [B]  time = 4.27447, size = 1065, normalized size = 4.53 \[ -\frac{\left (-d-4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right ) \left (d+4 e x-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right ) \sqrt{-\frac{\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}} \left (d+4 e x+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right )}{\left (\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right ) \left (-d-4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right )}} \sqrt{\frac{3 d^2+\left (\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right ) d+4 e \left (\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right ) x-2 \sqrt{d^4-64 a e^3}-\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}} \sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}}{\left (\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right ) \left (-d-4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right )}} F\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right ) \left (d+4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right )}{\left (\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right ) \left (-d-4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right )}}\right )|\frac{\left (\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right )^2}{\left (\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right )^2}\right )}{2 e \left (\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}-\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right ) \sqrt{\frac{\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}} \left (-d-4 e x+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right )}{\left (\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}+\sqrt{3 d^2+2 \sqrt{d^4-64 a e^3}}\right ) \left (-d-4 e x+\sqrt{3 d^2-2 \sqrt{d^4-64 a e^3}}\right )}} \sqrt{8 e^3 x^4+8 d e^2 x^3-d^3 x+8 a e^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4],x]

[Out]

-((-d + Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - 4*e*x)*(d - Sqrt[3*d^2 + 2*Sqrt[d
^4 - 64*a*e^3]] + 4*e*x)*Sqrt[-((Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]*(d + Sqrt[
3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]] + 4*e*x))/((Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]
- Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*(-d + Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^
3]] - 4*e*x)))]*Sqrt[(3*d^2 - 2*Sqrt[d^4 - 64*a*e^3] - Sqrt[3*d^2 - 2*Sqrt[d^4 -
64*a*e^3]]*Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]] + d*(Sqrt[3*d^2 - 2*Sqrt[d^4 -
64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]]) + 4*e*(Sqrt[3*d^2 - 2*Sqrt[d^
4 - 64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*x)/((Sqrt[3*d^2 - 2*Sqrt[
d^4 - 64*a*e^3]] + Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*(-d + Sqrt[3*d^2 - 2*Sq
rt[d^4 - 64*a*e^3]] - 4*e*x))]*EllipticF[ArcSin[Sqrt[((Sqrt[3*d^2 - 2*Sqrt[d^4 -
64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])*(d + Sqrt[3*d^2 - 2*Sqrt[d^4
- 64*a*e^3]] + 4*e*x))/((Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] + Sqrt[3*d^2 + 2*
Sqrt[d^4 - 64*a*e^3]])*(-d + Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - 4*e*x))]], (
Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] + Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])^2/(
Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]])^2])
/(2*e*(Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3
]])*Sqrt[(Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]*(-d + Sqrt[3*d^2 + 2*Sqrt[d^4 - 6
4*a*e^3]] - 4*e*x))/((Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] + Sqrt[3*d^2 + 2*Sqrt
[d^4 - 64*a*e^3]])*(-d + Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]] - 4*e*x))]*Sqrt[8*
a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4])

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Maple [B]  time = 0.045, size = 1704, normalized size = 7.3 \[ \text{result too large to display} \]

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)^(1/2),x)

[Out]

1/2*(1/4*(-d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2+1/4*(d*e+(3*d^
2*e^2-2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)*((-1/4*(d*e+(3*d^2*e^2-2*(-64*a*e
^3+d^4)^(1/2)*e^2)^(1/2))/e^2+1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(
1/2))/e^2)*(x-1/4*(-d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)/(-1/
4*(d*e+(3*d^2*e^2-2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2-1/4*(-d*e+(3*d^2*e^2+2
*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)/(x+1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)
^(1/2)*e^2)^(1/2))/e^2))^(1/2)*(x+1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^
2)^(1/2))/e^2)^2*((-1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2-
1/4*(-d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)*(x-1/4*(-d*e+(3*d^
2*e^2-2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)/(1/4*(-d*e+(3*d^2*e^2-2*(-64*a*e^
3+d^4)^(1/2)*e^2)^(1/2))/e^2-1/4*(-d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(
1/2))/e^2)/(x+1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2))^(1/2
)*((-1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2-1/4*(-d*e+(3*d^
2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)*(x+1/4*(d*e+(3*d^2*e^2-2*(-64*a*e
^3+d^4)^(1/2)*e^2)^(1/2))/e^2)/(-1/4*(d*e+(3*d^2*e^2-2*(-64*a*e^3+d^4)^(1/2)*e^2
)^(1/2))/e^2-1/4*(-d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)/(x+1/
4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2))^(1/2)/(-1/4*(d*e+(3*
d^2*e^2-2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2+1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3
+d^4)^(1/2)*e^2)^(1/2))/e^2)/(-1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^
(1/2))/e^2-1/4*(-d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)*2^(1/2)
/(e^3*(x-1/4*(-d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)*(x+1/4*(d
*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)*(x-1/4*(-d*e+(3*d^2*e^2-2
*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)*(x+1/4*(d*e+(3*d^2*e^2-2*(-64*a*e^3+d^4)
^(1/2)*e^2)^(1/2))/e^2))^(1/2)*EllipticF(((-1/4*(d*e+(3*d^2*e^2-2*(-64*a*e^3+d^4
)^(1/2)*e^2)^(1/2))/e^2+1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/
e^2)*(x-1/4*(-d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)/(-1/4*(d*e
+(3*d^2*e^2-2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2-1/4*(-d*e+(3*d^2*e^2+2*(-64*
a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)/(x+1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)
*e^2)^(1/2))/e^2))^(1/2),((-1/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/
2))/e^2-1/4*(-d*e+(3*d^2*e^2-2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)*(1/4*(-d*e
+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2+1/4*(d*e+(3*d^2*e^2-2*(-64*a
*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)/(1/4*(-d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e
^2)^(1/2))/e^2-1/4*(-d*e+(3*d^2*e^2-2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2)/(-1
/4*(d*e+(3*d^2*e^2+2*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2+1/4*(d*e+(3*d^2*e^2-2
*(-64*a*e^3+d^4)^(1/2)*e^2)^(1/2))/e^2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{8 a e^{2} - d^{3} x + 8 d e^{2} x^{3} + 8 e^{3} x^{4}}}\, dx \]

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)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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