3.43 \(\int \frac{x^4 \left (d+e x^4\right )}{a+b x^4+c x^8} \, dx\)

Optimal. Leaf size=433 \[ -\frac{\left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{e x}{c} \]

[Out]

(e*x)/c - ((c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(
1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*c^(5/4)*(-b - Sqrt[b
^2 - 4*a*c])^(3/4)) - ((c*d - b*e - (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])
*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*c^(5/4)*
(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - ((c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b
^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(
1/4)*c^(5/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - ((c*d - b*e - (b*c*d - b^2*e + 2*
a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(
1/4)])/(2*2^(1/4)*c^(5/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

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Rubi [A]  time = 2.40237, antiderivative size = 433, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\left (-\frac{2 a c e+b^2 (-e)+b c d}{\sqrt{b^2-4 a c}}-b e+c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{e x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(d + e*x^4))/(a + b*x^4 + c*x^8),x]

[Out]

(e*x)/c - ((c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(
1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*c^(5/4)*(-b - Sqrt[b
^2 - 4*a*c])^(3/4)) - ((c*d - b*e - (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])
*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*c^(5/4)*
(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - ((c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b
^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(
1/4)*c^(5/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) - ((c*d - b*e - (b*c*d - b^2*e + 2*
a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(
1/4)])/(2*2^(1/4)*c^(5/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

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Rubi in Sympy [A]  time = 169.92, size = 432, normalized size = 1. \[ \frac{e x}{c} - \frac{2^{\frac{3}{4}} \left (- 2 a c e + b \left (b e - c d\right ) - \sqrt{- 4 a c + b^{2}} \left (b e - c d\right )\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{5}{4}} \left (- b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{3}{4}} \left (- 2 a c e + b \left (b e - c d\right ) - \sqrt{- 4 a c + b^{2}} \left (b e - c d\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{5}{4}} \left (- b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{3}{4}} \left (- 2 a c e + b \left (b e - c d\right ) + \sqrt{- 4 a c + b^{2}} \left (b e - c d\right )\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{5}{4}} \left (- b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{3}{4}} \left (- 2 a c e + b \left (b e - c d\right ) + \sqrt{- 4 a c + b^{2}} \left (b e - c d\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{5}{4}} \left (- b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(e*x**4+d)/(c*x**8+b*x**4+a),x)

[Out]

e*x/c - 2**(3/4)*(-2*a*c*e + b*(b*e - c*d) - sqrt(-4*a*c + b**2)*(b*e - c*d))*at
an(2**(1/4)*c**(1/4)*x/(-b + sqrt(-4*a*c + b**2))**(1/4))/(4*c**(5/4)*(-b + sqrt
(-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b**2)) - 2**(3/4)*(-2*a*c*e + b*(b*e - c*d
) - sqrt(-4*a*c + b**2)*(b*e - c*d))*atanh(2**(1/4)*c**(1/4)*x/(-b + sqrt(-4*a*c
+ b**2))**(1/4))/(4*c**(5/4)*(-b + sqrt(-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b*
*2)) + 2**(3/4)*(-2*a*c*e + b*(b*e - c*d) + sqrt(-4*a*c + b**2)*(b*e - c*d))*ata
n(2**(1/4)*c**(1/4)*x/(-b - sqrt(-4*a*c + b**2))**(1/4))/(4*c**(5/4)*(-b - sqrt(
-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b**2)) + 2**(3/4)*(-2*a*c*e + b*(b*e - c*d)
+ sqrt(-4*a*c + b**2)*(b*e - c*d))*atanh(2**(1/4)*c**(1/4)*x/(-b - sqrt(-4*a*c
+ b**2))**(1/4))/(4*c**(5/4)*(-b - sqrt(-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b**
2))

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Mathematica [C]  time = 0.104407, size = 88, normalized size = 0.2 \[ \frac{e x}{c}-\frac{\text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b e \log (x-\text{$\#$1})+\text{$\#$1}^4 (-c) d \log (x-\text{$\#$1})+a e \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right ]}{4 c} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(d + e*x^4))/(a + b*x^4 + c*x^8),x]

[Out]

(e*x)/c - RootSum[a + b*#1^4 + c*#1^8 & , (a*e*Log[x - #1] - c*d*Log[x - #1]*#1^
4 + b*e*Log[x - #1]*#1^4)/(b*#1^3 + 2*c*#1^7) & ]/(4*c)

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Maple [C]  time = 0.007, size = 67, normalized size = 0.2 \[{\frac{ex}{c}}+{\frac{1}{4\,c}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+{{\it \_Z}}^{4}b+a \right ) }{\frac{ \left ( \left ( -be+cd \right ){{\it \_R}}^{4}-ae \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(e*x^4+d)/(c*x^8+b*x^4+a),x)

[Out]

e*x/c+1/4/c*sum(((-b*e+c*d)*_R^4-a*e)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*
c+_Z^4*b+a))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*x/c - integrate(-((c*d - b*e)*x^4 - a*e)/(c*x^8 + b*x^4 + a), x)/c

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Fricas [A]  time = 4.4671, size = 17662, normalized size = 40.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(4*c*sqrt(sqrt(1/2)*sqrt(-(b*c^4*d^4 - 4*(b^2*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*
c^2 - 3*a*b*c^3)*d^2*e^2 - 4*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*
a*b^3*c + 5*a^2*b*c^2)*e^4 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((c^8*d^8
- 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^
5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a
*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4
- 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e
^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10
- 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*
a^2*c^7)))*arctan(-1/2*((b^2*c^4 - 4*a*c^5)*d^4*e - 4*(b^3*c^3 - 4*a*b*c^4)*d^3*
e^2 + 6*(b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d^2*e^3 - 4*(b^5*c - 6*a*b^3*c^2 + 8
*a^2*b*c^3)*d*e^4 + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^5 + (2*(b^4
*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d - (b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*e)*s
qrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 -
8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b
^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28
*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a
^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e
^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(sqrt(1/2)*
sqrt(-(b*c^4*d^4 - 4*(b^2*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2
- 4*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e
^4 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b
^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 -
71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)
*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8
*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c +
11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^
2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))/((c^5*d^6 - 5
*b*c^4*d^5*e + 5*(2*b^2*c^3 - a*c^4)*d^4*e^2 - 10*(b^3*c^2 - a*b*c^3)*d^3*e^3 +
5*(b^4*c - a*b^2*c^2 - a^2*c^3)*d^2*e^4 - (b^5 + a*b^3*c - 7*a^2*b*c^2)*d*e^5 +
(a*b^4 - 3*a^2*b^2*c + a^3*c^2)*e^6)*x + sqrt(1/2)*(c^5*d^6 - 5*b*c^4*d^5*e + 5*
(2*b^2*c^3 - a*c^4)*d^4*e^2 - 10*(b^3*c^2 - a*b*c^3)*d^3*e^3 + 5*(b^4*c - a*b^2*
c^2 - a^2*c^3)*d^2*e^4 - (b^5 + a*b^3*c - 7*a^2*b*c^2)*d*e^5 + (a*b^4 - 3*a^2*b^
2*c + a^3*c^2)*e^6)*sqrt((2*(c^6*d^8 - 6*b*c^5*d^7*e + (15*b^2*c^4 - 4*a*c^5)*d^
6*e^2 - 10*(2*b^3*c^3 - a*b*c^4)*d^5*e^3 + 5*(3*b^4*c^2 - a*b^2*c^3 - 2*a^2*c^4)
*d^4*e^4 - 2*(3*b^5*c + 3*a*b^3*c^2 - 11*a^2*b*c^3)*d^3*e^5 + (b^6 + 7*a*b^4*c -
15*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^6 - 2*(a*b^5 - a^2*b^3*c - 3*a^3*b*c^2)*d*e^7
+ (a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*e^8)*x^2 + sqrt(1/2)*(2*(b^2*c^6 - 4*a*c^7)
*d^6 - 10*(b^3*c^5 - 4*a*b*c^6)*d^5*e + 7*(3*b^4*c^4 - 14*a*b^2*c^5 + 8*a^2*c^6)
*d^4*e^2 - 12*(2*b^5*c^3 - 11*a*b^3*c^4 + 12*a^2*b*c^5)*d^3*e^3 + 2*(8*b^6*c^2 -
52*a*b^4*c^3 + 87*a^2*b^2*c^4 - 28*a^3*c^5)*d^2*e^4 - 2*(3*b^7*c - 23*a*b^5*c^2
+ 53*a^2*b^3*c^3 - 36*a^3*b*c^4)*d*e^5 + (b^8 - 9*a*b^6*c + 27*a^2*b^4*c^2 - 30
*a^3*b^2*c^3 + 8*a^4*c^4)*e^6 + (2*(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64
*a^3*c^9)*d*e - (b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*e^2)*sq
rt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8
*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^
5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*
a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^
3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^
8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(-(b*c^4*d^4
- 4*(b^2*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2 - 4*(b^4*c - 4*
a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^4 + (b^4*c^5 -
8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7
)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 1
9*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*
b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^
5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 -
6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*
a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))/(c^6*d^8 - 6*b*c^5*d^7*e + (1
5*b^2*c^4 - 4*a*c^5)*d^6*e^2 - 10*(2*b^3*c^3 - a*b*c^4)*d^5*e^3 + 5*(3*b^4*c^2 -
a*b^2*c^3 - 2*a^2*c^4)*d^4*e^4 - 2*(3*b^5*c + 3*a*b^3*c^2 - 11*a^2*b*c^3)*d^3*e
^5 + (b^6 + 7*a*b^4*c - 15*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^6 - 2*(a*b^5 - a^2*b^3
*c - 3*a^3*b*c^2)*d*e^7 + (a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*e^8)))) - 4*c*sqrt(s
qrt(1/2)*sqrt(-(b*c^4*d^4 - 4*(b^2*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*a*b*c^3
)*d^2*e^2 - 4*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2
*b*c^2)*e^4 - (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e
+ 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b
^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a
^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^
2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a
*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^1
1 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))*arct
an(1/2*((b^2*c^4 - 4*a*c^5)*d^4*e - 4*(b^3*c^3 - 4*a*b*c^4)*d^3*e^2 + 6*(b^4*c^2
- 5*a*b^2*c^3 + 4*a^2*c^4)*d^2*e^3 - 4*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e^
4 + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^5 - (2*(b^4*c^6 - 8*a*b^2*c
^7 + 16*a^2*c^8)*d - (b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*e)*sqrt((c^8*d^8 - 8
*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e
^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^
3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3
*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7
+ (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 -
12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(sqrt(1/2)*sqrt(-(b*c^4*d^4
- 4*(b^2*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2 - 4*(b^4*c - 4*
a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^4 - (b^4*c^5 -
8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7
)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 1
9*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*
b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^
5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 -
6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*
a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))/((c^5*d^6 - 5*b*c^4*d^5*e + 5
*(2*b^2*c^3 - a*c^4)*d^4*e^2 - 10*(b^3*c^2 - a*b*c^3)*d^3*e^3 + 5*(b^4*c - a*b^2
*c^2 - a^2*c^3)*d^2*e^4 - (b^5 + a*b^3*c - 7*a^2*b*c^2)*d*e^5 + (a*b^4 - 3*a^2*b
^2*c + a^3*c^2)*e^6)*x + sqrt(1/2)*(c^5*d^6 - 5*b*c^4*d^5*e + 5*(2*b^2*c^3 - a*c
^4)*d^4*e^2 - 10*(b^3*c^2 - a*b*c^3)*d^3*e^3 + 5*(b^4*c - a*b^2*c^2 - a^2*c^3)*d
^2*e^4 - (b^5 + a*b^3*c - 7*a^2*b*c^2)*d*e^5 + (a*b^4 - 3*a^2*b^2*c + a^3*c^2)*e
^6)*sqrt((2*(c^6*d^8 - 6*b*c^5*d^7*e + (15*b^2*c^4 - 4*a*c^5)*d^6*e^2 - 10*(2*b^
3*c^3 - a*b*c^4)*d^5*e^3 + 5*(3*b^4*c^2 - a*b^2*c^3 - 2*a^2*c^4)*d^4*e^4 - 2*(3*
b^5*c + 3*a*b^3*c^2 - 11*a^2*b*c^3)*d^3*e^5 + (b^6 + 7*a*b^4*c - 15*a^2*b^2*c^2
- 4*a^3*c^3)*d^2*e^6 - 2*(a*b^5 - a^2*b^3*c - 3*a^3*b*c^2)*d*e^7 + (a^2*b^4 - 3*
a^3*b^2*c + a^4*c^2)*e^8)*x^2 + sqrt(1/2)*(2*(b^2*c^6 - 4*a*c^7)*d^6 - 10*(b^3*c
^5 - 4*a*b*c^6)*d^5*e + 7*(3*b^4*c^4 - 14*a*b^2*c^5 + 8*a^2*c^6)*d^4*e^2 - 12*(2
*b^5*c^3 - 11*a*b^3*c^4 + 12*a^2*b*c^5)*d^3*e^3 + 2*(8*b^6*c^2 - 52*a*b^4*c^3 +
87*a^2*b^2*c^4 - 28*a^3*c^5)*d^2*e^4 - 2*(3*b^7*c - 23*a*b^5*c^2 + 53*a^2*b^3*c^
3 - 36*a^3*b*c^4)*d*e^5 + (b^8 - 9*a*b^6*c + 27*a^2*b^4*c^2 - 30*a^3*b^2*c^3 + 8
*a^4*c^4)*e^6 - (2*(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*d*e -
(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*e^2)*sqrt((c^8*d^8 - 8*
b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^
3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3
*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*
a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 +
(b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 1
2*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(-(b*c^4*d^4 - 4*(b^2*c^3 -
2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2 - 4*(b^4*c - 4*a*b^2*c^2 + 2*a^
2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^4 - (b^4*c^5 - 8*a*b^2*c^6 + 16
*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7
*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e
^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b
^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^
3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 +
a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4
*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))/(c^6*d^8 - 6*b*c^5*d^7*e + (15*b^2*c^4 - 4*a*
c^5)*d^6*e^2 - 10*(2*b^3*c^3 - a*b*c^4)*d^5*e^3 + 5*(3*b^4*c^2 - a*b^2*c^3 - 2*a
^2*c^4)*d^4*e^4 - 2*(3*b^5*c + 3*a*b^3*c^2 - 11*a^2*b*c^3)*d^3*e^5 + (b^6 + 7*a*
b^4*c - 15*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^6 - 2*(a*b^5 - a^2*b^3*c - 3*a^3*b*c^2
)*d*e^7 + (a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*e^8)))) + c*sqrt(sqrt(1/2)*sqrt(-(b*
c^4*d^4 - 4*(b^2*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2 - 4*(b^4
*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^4 + (b^4
*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 -
3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*
c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5
+ 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c -
5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^
4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^1
2 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))*log((c^5*d^6 - 5*b*c^4
*d^5*e + 5*(2*b^2*c^3 - a*c^4)*d^4*e^2 - 10*(b^3*c^2 - a*b*c^3)*d^3*e^3 + 5*(b^4
*c - a*b^2*c^2 - a^2*c^3)*d^2*e^4 - (b^5 + a*b^3*c - 7*a^2*b*c^2)*d*e^5 + (a*b^4
- 3*a^2*b^2*c + a^3*c^2)*e^6)*x + 1/2*((b^2*c^4 - 4*a*c^5)*d^4*e - 4*(b^3*c^3 -
4*a*b*c^4)*d^3*e^2 + 6*(b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d^2*e^3 - 4*(b^5*c -
6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e^4 + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^
3)*e^5 + (2*(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d - (b^5*c^5 - 8*a*b^3*c^6 + 16
*a^2*b*c^7)*e)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 -
8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*
d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 2
8*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a
^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*
c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))
*sqrt(sqrt(1/2)*sqrt(-(b*c^4*d^4 - 4*(b^2*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*
a*b*c^3)*d^2*e^2 - 4*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c
+ 5*a^2*b*c^2)*e^4 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^
7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 +
2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4
+ 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*
c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^
8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*
b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)
))) - c*sqrt(sqrt(1/2)*sqrt(-(b*c^4*d^4 - 4*(b^2*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*c
^2 - 3*a*b*c^3)*d^2*e^2 - 4*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a
*b^3*c + 5*a^2*b*c^2)*e^4 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((c^8*d^8 -
8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5
*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*
b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 -
3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^
7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10
- 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a
^2*c^7)))*log((c^5*d^6 - 5*b*c^4*d^5*e + 5*(2*b^2*c^3 - a*c^4)*d^4*e^2 - 10*(b^3
*c^2 - a*b*c^3)*d^3*e^3 + 5*(b^4*c - a*b^2*c^2 - a^2*c^3)*d^2*e^4 - (b^5 + a*b^3
*c - 7*a^2*b*c^2)*d*e^5 + (a*b^4 - 3*a^2*b^2*c + a^3*c^2)*e^6)*x - 1/2*((b^2*c^4
- 4*a*c^5)*d^4*e - 4*(b^3*c^3 - 4*a*b*c^4)*d^3*e^2 + 6*(b^4*c^2 - 5*a*b^2*c^3 +
4*a^2*c^4)*d^2*e^3 - 4*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e^4 + (b^6 - 7*a*b
^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^5 + (2*(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)
*d - (b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*e)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4
*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c
^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b
*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^
6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6
*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 +
48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(sqrt(1/2)*sqrt(-(b*c^4*d^4 - 4*(b^2*c^3 -
2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2 - 4*(b^4*c - 4*a*b^2*c^2 + 2*a^
2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^4 + (b^4*c^5 - 8*a*b^2*c^6 + 16
*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7
*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e
^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b
^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^
3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 +
a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4
*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))) + c*sqrt(sqrt(1/2)*sqrt(-(b*c^4*d^4 - 4*(b^2
*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2 - 4*(b^4*c - 4*a*b^2*c^2
+ 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^4 - (b^4*c^5 - 8*a*b^2*c
^6 + 16*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2
- 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6
)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 -
28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7
*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^
2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)
))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))*log((c^5*d^6 - 5*b*c^4*d^5*e + 5*(2*b^
2*c^3 - a*c^4)*d^4*e^2 - 10*(b^3*c^2 - a*b*c^3)*d^3*e^3 + 5*(b^4*c - a*b^2*c^2 -
a^2*c^3)*d^2*e^4 - (b^5 + a*b^3*c - 7*a^2*b*c^2)*d*e^5 + (a*b^4 - 3*a^2*b^2*c +
a^3*c^2)*e^6)*x + 1/2*((b^2*c^4 - 4*a*c^5)*d^4*e - 4*(b^3*c^3 - 4*a*b*c^4)*d^3*
e^2 + 6*(b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d^2*e^3 - 4*(b^5*c - 6*a*b^3*c^2 + 8
*a^2*b*c^3)*d*e^4 + (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^5 - (2*(b^4
*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d - (b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*e)*s
qrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 -
8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b
^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28
*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a
^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e
^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(sqrt(1/2)*
sqrt(-(b*c^4*d^4 - 4*(b^2*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2
- 4*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e
^4 - (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b
^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 -
71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)
*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8
*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c +
11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^
2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))) - c*sqrt(sqr
t(1/2)*sqrt(-(b*c^4*d^4 - 4*(b^2*c^3 - 2*a*c^4)*d^3*e + 6*(b^3*c^2 - 3*a*b*c^3)*
d^2*e^2 - 4*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 5*a*b^3*c + 5*a^2*b
*c^2)*e^4 - (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e +
4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4
*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2
*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*
e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b
^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11
+ 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))*log((c
^5*d^6 - 5*b*c^4*d^5*e + 5*(2*b^2*c^3 - a*c^4)*d^4*e^2 - 10*(b^3*c^2 - a*b*c^3)*
d^3*e^3 + 5*(b^4*c - a*b^2*c^2 - a^2*c^3)*d^2*e^4 - (b^5 + a*b^3*c - 7*a^2*b*c^2
)*d*e^5 + (a*b^4 - 3*a^2*b^2*c + a^3*c^2)*e^6)*x - 1/2*((b^2*c^4 - 4*a*c^5)*d^4*
e - 4*(b^3*c^3 - 4*a*b*c^4)*d^3*e^2 + 6*(b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d^2*
e^3 - 4*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e^4 + (b^6 - 7*a*b^4*c + 13*a^2*b^
2*c^2 - 4*a^3*c^3)*e^5 - (2*(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d - (b^5*c^5 -
8*a*b^3*c^6 + 16*a^2*b*c^7)*e)*sqrt((c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*
a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^
5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 +
4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5
*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*
c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12
- 64*a^3*c^13)))*sqrt(sqrt(1/2)*sqrt(-(b*c^4*d^4 - 4*(b^2*c^3 - 2*a*c^4)*d^3*e +
6*(b^3*c^2 - 3*a*b*c^3)*d^2*e^2 - 4*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (
b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^4 - (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt((
c^8*d^8 - 8*b*c^7*d^7*e + 4*(7*b^2*c^6 - 3*a*c^7)*d^6*e^2 - 8*(7*b^3*c^5 - 8*a*b
*c^6)*d^5*e^3 + 2*(35*b^4*c^4 - 71*a*b^2*c^5 + 19*a^2*c^6)*d^4*e^4 - 8*(7*b^5*c^
3 - 21*a*b^3*c^4 + 13*a^2*b*c^5)*d^3*e^5 + 4*(7*b^6*c^2 - 28*a*b^4*c^3 + 28*a^2*
b^2*c^4 - 3*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*
c^4)*d*e^7 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^8)/(
b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c
^6 + 16*a^2*c^7)))) + 4*e*x)/c

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(e*x**4+d)/(c*x**8+b*x**4+a),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x^4 + d)*x^4/(c*x^8 + b*x^4 + a), x)