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

Optimal. Leaf size=663 \[ \frac{1}{3} \left (\frac{d}{4 e}+x\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}-\frac{2 d^2 \left (\frac{d}{4 e}+x\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{\sqrt{256 a e^3+5 d^4} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )}+\frac{\sqrt [4]{256 a e^3+5 d^4} \left (-3 d^2 \sqrt{256 a e^3+5 d^4}+256 a e^3+5 d^4\right ) \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{48 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac{d^2 \left (256 a e^3+5 d^4\right )^{3/4} \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{8 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \]

[Out]

((d/(4*e) + x)*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4])/3 - (2*d^2*(d/(4
*e) + x)*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4])/(Sqrt[5*d^4 + 256*a*e^
3]*(1 + (16*e^2*(d/(4*e) + x)^2)/Sqrt[5*d^4 + 256*a*e^3])) + (d^2*(5*d^4 + 256*a
*e^3)^(3/4)*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])*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])/(8*Sqrt[2]*e^2*Sqrt
[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)^(1/4)*(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])*Ellipt
icF[2*ArcTan[(d + 4*e*x)/(5*d^4 + 256*a*e^3)^(1/4)], (1 + (3*d^2)/Sqrt[5*d^4 + 2
56*a*e^3])/2])/(48*Sqrt[2]*e^2*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 = 1.57782, antiderivative size = 663, 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 \[ \frac{(d+4 e x) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{12 e}-\frac{d^2 (d+4 e x) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{2 e \sqrt{256 a e^3+5 d^4} \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right )}+\frac{\sqrt [4]{256 a e^3+5 d^4} \left (-3 d^2 \sqrt{256 a e^3+5 d^4}+256 a e^3+5 d^4\right ) \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{48 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac{d^2 \left (256 a e^3+5 d^4\right )^{3/4} \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{(d+4 e x)^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{8 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((d + 4*e*x)*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4])/(12*e) - (d^2*(d +
4*e*x)*Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4])/(2*e*Sqrt[5*d^4 + 256*a
*e^3]*(1 + (d + 4*e*x)^2/Sqrt[5*d^4 + 256*a*e^3])) + (d^2*(5*d^4 + 256*a*e^3)^(3
/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 + 25
6*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])/(8*Sqrt[2]*e^2*Sqrt[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)^(1/4)*(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 + 2
56*a*e^3)*(1 + (d + 4*e*x)^2/Sqrt[5*d^4 + 256*a*e^3])^2)]*(1 + (d + 4*e*x)^2/Sqr
t[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])/(48*Sqrt[2]*e^2*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 = 139.962, size = 673, normalized size = 1.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**2*(d/(4*e) + x)*sqrt(-192*d**2*e*(d/(4*e) + x)**2 + 512*e**3*(d/(4*e) + x)**
4 + 2*(256*a*e**3 + 5*d**4)/e)/(4*sqrt(256*a*e**3 + 5*d**4)*(16*e**2*(d/(4*e) +
x)**2/sqrt(256*a*e**3 + 5*d**4) + 1)) + sqrt(2)*d**2*sqrt((512*a*e**3 + 10*d**4
- 192*d**2*e**2*(d/(4*e) + x)**2 + 512*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)**(3/4)*(16*e**2*(d/(4*e) + x)**2/sqrt(256*a*e**3 + 5*d**4) + 1)*ellipti
c_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)/(16*e**2*sqrt(-192*d**2*e*(d/(4*e) + x)**2 + 512*e**3*(d
/(4*e) + x)**4 + 2*(256*a*e**3 + 5*d**4)/e)) + (d/(4*e) + x)*sqrt(-192*d**2*e*(d
/(4*e) + x)**2 + 512*e**3*(d/(4*e) + x)**4 + 2*(256*a*e**3 + 5*d**4)/e)/24 - sqr
t(2)*sqrt((512*a*e**3 + 10*d**4 - 192*d**2*e**2*(d/(4*e) + x)**2 + 512*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))*(3*d**2 - sqrt(256*a*e**3 + 5*d**4))*(256*a*e**3 + 5*d**4)**(
3/4)*(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)/(96*e**2*sqrt(-192*d**2*e*(d/(4*e) + x)**2 + 512*e**3*(d/(4*e) + x)
**4 + 2*(256*a*e**3 + 5*d**4)/e))

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Mathematica [B]  time = 6.19812, size = 7543, normalized size = 11.38 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [B]  time = 0.283, size = 7887, normalized size = 11.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(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. \[{\rm integral}\left (\sqrt{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(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. \[ \int \sqrt{8 a e^{2} - d^{3} x + 8 d e^{2} x^{3} + 8 e^{3} x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(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. \[ \int \sqrt{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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