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

Optimal. Leaf size=748 \[ \frac{4 e \left (\frac{d}{4 e}+x\right ) \left (-256 a e^3+13 d^4-48 d^2 e^2 \left (\frac{d}{4 e}+x\right )^2\right )}{\left (-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac{384 d^2 e^2 \left (\frac{d}{4 e}+x\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{\left (d^4-64 a e^3\right ) \left (256 a e^3+5 d^4\right )^{3/2} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )}-\frac{2 \sqrt{2} \left (-3 d^2 \sqrt{256 a e^3+5 d^4}+256 a e^3+5 d^4\right ) \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{\left (d^4-64 a e^3\right ) \left (256 a e^3+5 d^4\right )^{3/4} \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}-\frac{12 \sqrt{2} d^2 \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{\left (d^4-64 a e^3\right ) \sqrt [4]{256 a e^3+5 d^4} \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \]

[Out]

(4*e*(d/(4*e) + x)*(13*d^4 - 256*a*e^3 - 48*d^2*e^2*(d/(4*e) + x)^2))/((5*d^8 -
64*a*d^4*e^3 - 16384*a^2*e^6)*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4]) +
(384*d^2*e^2*(d/(4*e) + x)*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4])/((d
^4 - 64*a*e^3)*(5*d^4 + 256*a*e^3)^(3/2)*(1 + (16*e^2*(d/(4*e) + x)^2)/Sqrt[5*d^
4 + 256*a*e^3])) - (12*Sqrt[2]*d^2*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])*EllipticE[2*ArcTa
n[(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])/((d^4 - 64*a*e^3)*(5*d^4 + 256*a*e^3)^(1/4)*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x
^3 + 8*e^3*x^4]) - (2*Sqrt[2]*(5*d^4 + 256*a*e^3 - 3*d^2*Sqrt[5*d^4 + 256*a*e^3]
)*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])/((d^4 - 64*a*e^3)*(5*d^4 + 25
6*a*e^3)^(3/4)*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 = 1.77893, antiderivative size = 748, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147 \[ \frac{(d+4 e x) \left (-256 a e^3+13 d^4-3 d^2 (d+4 e x)^2\right )}{\left (-16384 a^2 e^6-64 a d^4 e^3+5 d^8\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac{96 d^2 e (d+4 e x) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{\left (d^4-64 a e^3\right ) \left (256 a e^3+5 d^4\right )^{3/2} \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right )}-\frac{2 \sqrt{2} \left (-3 d^2 \sqrt{256 a e^3+5 d^4}+256 a e^3+5 d^4\right ) \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{\left (d^4-64 a e^3\right ) \left (256 a e^3+5 d^4\right )^{3/4} \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}-\frac{12 \sqrt{2} d^2 \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{\left (d^4-64 a e^3\right ) \sqrt [4]{256 a e^3+5 d^4} \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((d + 4*e*x)*(13*d^4 - 256*a*e^3 - 3*d^2*(d + 4*e*x)^2))/((5*d^8 - 64*a*d^4*e^3
- 16384*a^2*e^6)*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4]) + (96*d^2*e*(d
+ 4*e*x)*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4])/((d^4 - 64*a*e^3)*(5*
d^4 + 256*a*e^3)^(3/2)*(1 + (d + 4*e*x)^2/Sqrt[5*d^4 + 256*a*e^3])) - (12*Sqrt[2
]*d^2*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])*EllipticE[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])/((d^4 - 64*a*e^3)*(5*d^4 + 256*a*e^3)^(1/4)*Sqr
t[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4]) - (2*Sqrt[2]*(5*d^4 + 256*a*e^3 -
3*d^2*Sqrt[5*d^4 + 256*a*e^3])*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])/((d^4 - 64*a*e^3)*(5*d
^4 + 256*a*e^3)^(3/4)*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 = 157.726, size = 777, normalized size = 1.04 \[ \text{result too large to display} \]

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)**(3/2),x)

[Out]

48*sqrt(2)*d**2*e**2*(d/(4*e) + x)*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)/((-64*a*e**3 + d**4)*(256*a*e**3 + 5*
d**4)**(3/2)*(16*e**2*(d/(4*e) + x)**2/sqrt(256*a*e**3 + 5*d**4) + 1)) - 12*sqrt
(2)*d**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))*(16*e**2*(d/(4*e) + x)**2/sqrt(256*a*e**3 + 5*d**4) + 1)*el
liptic_e(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)/((-64*a*e**3 + d**4)*(256*a*e**3 + 5*d**4)**(1/4)*s
qrt(-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)) + 128*sqrt(2)*e*(d/(4*e) + x)*(-131072*a*e**3 + 6656*d**4 - 24576*d**2*e
**2*(d/(4*e) + x)**2)/((-67108864*a**2*e**6 - 262144*a*d**4*e**3 + 20480*d**8)*s
qrt(-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)) - 2*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))*(16*e**2*(d/(4*e) + x)**2/sqrt(256*a*e**3 + 5*d*
*4) + 1)*(256*a*e**3 + 5*d**4 - 3*d**2*sqrt(256*a*e**3 + 5*d**4))*elliptic_f(2*a
tan(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)/((-64*a*e**3 + d**4)*(256*a*e**3 + 5*d**4)**(3/4)*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 = 6.23499, size = 7629, normalized size = 10.2 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.067, size = 8103, normalized size = 10.8 \[ \text{output too large to display} \]

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)^(3/2),x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)**(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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