3.157 \(\int \frac{1}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=864 \[ \frac{x e^4}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac{4 c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{\left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c x \left (c d^2-2 c e x^2 d-a e^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (c x^4+a\right )} \]

[Out]

(e^4*x)/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) + (c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^2)
)/(4*a*(c*d^2 + a*e^2)^2*(a + c*x^4)) + (4*c*Sqrt[d]*e^(7/2)*ArcTan[(Sqrt[e]*x)/
Sqrt[d]])/(c*d^2 + a*e^2)^3 + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*(
c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 - 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTa
n[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(
3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/
a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*e^2*(3*c*d^2 - 4*Sqrt
[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(
3/4)*(c*d^2 + a*e^2)^3) + (c^(3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*A
rcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) -
(c^(3/4)*e^2*(3*c*d^2 + 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(
1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(3/4)*
(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4
)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*e^2*(3*c*d
^2 + 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sq
rt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) + (c^(3/4)*(3*c*d^2 + 2*Sqrt[a
]*Sqrt[c]*d*e - 3*a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])
/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2)

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Rubi [A]  time = 1.6853, antiderivative size = 864, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{x e^4}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{2 d^{3/2} \left (c d^2+a e^2\right )^2}+\frac{4 c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) e^{7/2}}{\left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2-4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^2}{2 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}+\frac{c^{3/4} \left (3 c d^2+4 \sqrt{a} \sqrt{c} e d-a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^2}{4 \sqrt{2} a^{3/4} \left (c d^2+a e^2\right )^3}-\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2-2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}-\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c^{3/4} \left (3 c d^2+2 \sqrt{a} \sqrt{c} e d-3 a e^2\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^2+a e^2\right )^2}+\frac{c x \left (c d^2-2 c e x^2 d-a e^2\right )}{4 a \left (c d^2+a e^2\right )^2 \left (c x^4+a\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x^2)^2*(a + c*x^4)^2),x]

[Out]

(e^4*x)/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) + (c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^2)
)/(4*a*(c*d^2 + a*e^2)^2*(a + c*x^4)) + (4*c*Sqrt[d]*e^(7/2)*ArcTan[(Sqrt[e]*x)/
Sqrt[d]])/(c*d^2 + a*e^2)^3 + (e^(7/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*(
c*d^2 + a*e^2)^2) - (c^(3/4)*e^2*(3*c*d^2 - 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTa
n[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(
3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/
a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*e^2*(3*c*d^2 - 4*Sqrt
[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(
3/4)*(c*d^2 + a*e^2)^3) + (c^(3/4)*(3*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*A
rcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) -
(c^(3/4)*e^2*(3*c*d^2 + 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(
1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) - (c^(3/4)*
(3*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4
)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2) + (c^(3/4)*e^2*(3*c*d
^2 + 4*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sq
rt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^2 + a*e^2)^3) + (c^(3/4)*(3*c*d^2 + 2*Sqrt[a
]*Sqrt[c]*d*e - 3*a*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])
/(16*Sqrt[2]*a^(7/4)*(c*d^2 + a*e^2)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.1605, size = 540, normalized size = 0.62 \[ \frac{-\frac{\sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3-7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e+12 a c d^2 e^2+3 c^2 d^4\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}+\frac{\sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3-7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e+12 a c d^2 e^2+3 c^2 d^4\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}+\frac{2 \sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3+7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e-12 a c d^2 e^2-3 c^2 d^4\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac{2 \sqrt{2} c^{3/4} \left (18 a^{3/2} \sqrt{c} d e^3+7 a^2 e^4+2 \sqrt{a} c^{3/2} d^3 e-12 a c d^2 e^2-3 c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+\frac{8 c x \left (a e^2+c d^2\right ) \left (c d \left (d-2 e x^2\right )-a e^2\right )}{a \left (a+c x^4\right )}+\frac{16 e^4 x \left (a e^2+c d^2\right )}{d \left (d+e x^2\right )}+\frac{16 e^{7/2} \left (a e^2+9 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{3/2}}}{32 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x^2)^2*(a + c*x^4)^2),x]

[Out]

((16*e^4*(c*d^2 + a*e^2)*x)/(d*(d + e*x^2)) + (8*c*(c*d^2 + a*e^2)*x*(-(a*e^2) +
c*d*(d - 2*e*x^2)))/(a*(a + c*x^4)) + (16*e^(7/2)*(9*c*d^2 + a*e^2)*ArcTan[(Sqr
t[e]*x)/Sqrt[d]])/d^(3/2) + (2*Sqrt[2]*c^(3/4)*(-3*c^2*d^4 + 2*Sqrt[a]*c^(3/2)*d
^3*e - 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 + 7*a^2*e^4)*ArcTan[1 - (Sqrt[2
]*c^(1/4)*x)/a^(1/4)])/a^(7/4) - (2*Sqrt[2]*c^(3/4)*(-3*c^2*d^4 + 2*Sqrt[a]*c^(3
/2)*d^3*e - 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 + 7*a^2*e^4)*ArcTan[1 + (S
qrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) - (Sqrt[2]*c^(3/4)*(3*c^2*d^4 + 2*Sqrt[a]*c^
(3/2)*d^3*e + 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 - 7*a^2*e^4)*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (Sqrt[2]*c^(3/4)*(3*c^2*d
^4 + 2*Sqrt[a]*c^(3/2)*d^3*e + 12*a*c*d^2*e^2 + 18*a^(3/2)*Sqrt[c]*d*e^3 - 7*a^2
*e^4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4))/(32*(c*d^
2 + a*e^2)^3)

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Maple [A]  time = 0.026, size = 1169, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(a*e^2+c*d^2)^3*c^2/(c*x^4+a)*d*e^3*x^3-1/2/(a*e^2+c*d^2)^3*c^3/(c*x^4+a)*d
^3*e/a*x^3-1/4/(a*e^2+c*d^2)^3*c/(c*x^4+a)*a*x*e^4+1/4/(a*e^2+c*d^2)^3*c^3/(c*x^
4+a)/a*x*d^4-7/16/(a*e^2+c*d^2)^3*c*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)
^(1/4)*x+1)*e^4+3/4/(a*e^2+c*d^2)^3*c^2/a*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
1/c*a)^(1/4)*x+1)*d^2*e^2+3/16/(a*e^2+c*d^2)^3*c^3/a^2*(1/c*a)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^4-7/16/(a*e^2+c*d^2)^3*c*(1/c*a)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*e^4+3/4/(a*e^2+c*d^2)^3*c^2/a*(1/c*a)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^2*e^2+3/16/(a*e^2+c*d^2)^3*c^3/a^2*(1/
c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^4-7/32/(a*e^2+c*d^2)^3*c*
(1/c*a)^(1/4)*2^(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a
)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*e^4+3/8/(a*e^2+c*d^2)^3*c^2/a*(1/c*a)^(1/4)*2^
(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2
)+(1/c*a)^(1/2)))*d^2*e^2+3/32/(a*e^2+c*d^2)^3*c^3/a^2*(1/c*a)^(1/4)*2^(1/2)*ln(
(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)
^(1/2)))*d^4-9/16/(a*e^2+c*d^2)^3*c/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)^(1/4)*
x*2^(1/2)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d*e^3-1/16
/(a*e^2+c*d^2)^3*c^2/a/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/
c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d^3*e-9/8/(a*e^2+c*d^2)
^3*c/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d*e^3-1/8/(a*e^2+c*
d^2)^3*c^2/a/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^3*e-9/8/(
a*e^2+c*d^2)^3*c/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d*e^3-1
/8/(a*e^2+c*d^2)^3*c^2/a/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)
*d^3*e+1/2*e^6/(a*e^2+c*d^2)^3/d*x/(e*x^2+d)*a+1/2*e^4/(a*e^2+c*d^2)^3*d*x/(e*x^
2+d)*c+1/2*e^6/(a*e^2+c*d^2)^3/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a+9/2*e^4/(
a*e^2+c*d^2)^3*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 174.494, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(8*(c^3*d^4*e^2 - a^2*c*e^6)*x^5 + 4*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + a^2*c
*d*e^5)*x^3 - (a^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e^6 +
(a*c^4*d^7*e + 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^6 + (a*c^
4*d^8 + 3*a^2*c^3*d^6*e^2 + 3*a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6)*x^4 + (a^2*c^3*d^
7*e + 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x^2)*sqrt((12*c^5*d^7*e +
156*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 + (a^3*c^6*d^12 + 6
*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8
+ 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 516
4*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c
^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^
16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*
d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^
12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66
*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5
*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8
*c*d^2*e^10 + a^9*e^12))*log((81*c^7*d^10 + 1053*a*c^6*d^8*e^2 + 3602*a^2*c^5*d^
6*e^4 - 2958*a^3*c^4*d^4*e^6 - 23667*a^4*c^3*d^2*e^8 + 2401*a^5*c^2*e^10)*x + (2
7*a^2*c^7*d^12 + 312*a^3*c^6*d^10*e^2 + 843*a^4*c^5*d^8*e^4 - 1592*a^5*c^4*d^6*e
^6 - 5967*a^6*c^3*d^4*e^8 + 4032*a^7*c^2*d^2*e^10 - 343*a^8*c*e^12 + 2*(a^6*c^7*
d^15*e + 15*a^7*c^6*d^13*e^3 + 69*a^8*c^5*d^11*e^5 + 155*a^9*c^4*d^9*e^7 + 195*a
^10*c^3*d^7*e^9 + 141*a^11*c^2*d^5*e^11 + 55*a^12*c*d^3*e^13 + 9*a^13*d*e^15)*sq
rt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*
d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e
^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d
^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 +
792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*
a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2
*e^22 + a^19*e^24)))*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^
5 - 252*a^3*c^2*d*e^7 + (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4
+ 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(
81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*
e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 -
48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e
^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*
a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*
c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22
+ a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6
*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))) + (a^2*c^3*d^
8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^7*e + 3*a^2*c^3
*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^6 + (a*c^4*d^8 + 3*a^2*c^3*d^6*e^2
+ 3*a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6)*x^4 + (a^2*c^3*d^7*e + 3*a^3*c^2*d^5*e^3 +
3*a^4*c*d^3*e^5 + a^5*d*e^7)*x^2)*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*
a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 + (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^
5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9
*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776
*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*
c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a
^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d
^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^
14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a
^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^
8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))*
log((81*c^7*d^10 + 1053*a*c^6*d^8*e^2 + 3602*a^2*c^5*d^6*e^4 - 2958*a^3*c^4*d^4*
e^6 - 23667*a^4*c^3*d^2*e^8 + 2401*a^5*c^2*e^10)*x - (27*a^2*c^7*d^12 + 312*a^3*
c^6*d^10*e^2 + 843*a^4*c^5*d^8*e^4 - 1592*a^5*c^4*d^6*e^6 - 5967*a^6*c^3*d^4*e^8
+ 4032*a^7*c^2*d^2*e^10 - 343*a^8*c*e^12 + 2*(a^6*c^7*d^15*e + 15*a^7*c^6*d^13*
e^3 + 69*a^8*c^5*d^11*e^5 + 155*a^9*c^4*d^9*e^7 + 195*a^10*c^3*d^7*e^9 + 141*a^1
1*c^2*d^5*e^11 + 55*a^12*c*d^3*e^13 + 9*a^13*d*e^15)*sqrt(-(81*c^11*d^16 + 1224*
a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*
d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e
^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^2
0*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 +
924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a
^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))*sqrt
((12*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 + (
a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15
*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^1
0*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e
^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 +
2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4
+ 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*
a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c
^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*
d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2
*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))) - (a^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3
*a^4*c*d^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^7*e + 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*
e^5 + a^4*c*d*e^7)*x^6 + (a*c^4*d^8 + 3*a^2*c^3*d^6*e^2 + 3*a^3*c^2*d^4*e^4 + a^
4*c*d^2*e^6)*x^4 + (a^2*c^3*d^7*e + 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*
e^7)*x^2)*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3
*c^2*d*e^7 - (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^
3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^1
6 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130
*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*
c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9
*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^
14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^1
6 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^2
4)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^
6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))*log((81*c^7*d^10 + 1053*a
*c^6*d^8*e^2 + 3602*a^2*c^5*d^6*e^4 - 2958*a^3*c^4*d^4*e^6 - 23667*a^4*c^3*d^2*e
^8 + 2401*a^5*c^2*e^10)*x + (27*a^2*c^7*d^12 + 312*a^3*c^6*d^10*e^2 + 843*a^4*c^
5*d^8*e^4 - 1592*a^5*c^4*d^6*e^6 - 5967*a^6*c^3*d^4*e^8 + 4032*a^7*c^2*d^2*e^10
- 343*a^8*c*e^12 - 2*(a^6*c^7*d^15*e + 15*a^7*c^6*d^13*e^3 + 69*a^8*c^5*d^11*e^5
+ 155*a^9*c^4*d^9*e^7 + 195*a^10*c^3*d^7*e^9 + 141*a^11*c^2*d^5*e^11 + 55*a^12*
c*d^3*e^13 + 9*a^13*d*e^15)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^
2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d
^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/
(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18
*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 +
792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^1
7*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))*sqrt((12*c^5*d^7*e + 156*a*c^
4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 - (a^3*c^6*d^12 + 6*a^4*c^5*
d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*
c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9
*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^
10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*
c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6
+ 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*
a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2
*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2
+ 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^
10 + a^9*e^12))) + (a^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*
e^6 + (a*c^4*d^7*e + 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^6 +
(a*c^4*d^8 + 3*a^2*c^3*d^6*e^2 + 3*a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6)*x^4 + (a^2*c
^3*d^7*e + 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x^2)*sqrt((12*c^5*d^
7*e + 156*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 - (a^3*c^6*d^1
2 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^
4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2
+ 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*
a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c
^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10
*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^
12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18
+ 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^
4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 +
6*a^8*c*d^2*e^10 + a^9*e^12))*log((81*c^7*d^10 + 1053*a*c^6*d^8*e^2 + 3602*a^2*c
^5*d^6*e^4 - 2958*a^3*c^4*d^4*e^6 - 23667*a^4*c^3*d^2*e^8 + 2401*a^5*c^2*e^10)*x
- (27*a^2*c^7*d^12 + 312*a^3*c^6*d^10*e^2 + 843*a^4*c^5*d^8*e^4 - 1592*a^5*c^4*
d^6*e^6 - 5967*a^6*c^3*d^4*e^8 + 4032*a^7*c^2*d^2*e^10 - 343*a^8*c*e^12 - 2*(a^6
*c^7*d^15*e + 15*a^7*c^6*d^13*e^3 + 69*a^8*c^5*d^11*e^5 + 155*a^9*c^4*d^9*e^7 +
195*a^10*c^3*d^7*e^9 + 141*a^11*c^2*d^5*e^11 + 55*a^12*c*d^3*e^13 + 9*a^13*d*e^1
5)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3
*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*
d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c
^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*
e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 +
495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*
c*d^2*e^22 + a^19*e^24)))*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*a^2*c^3*d
^3*e^5 - 252*a^3*c^2*d*e^7 - (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8
*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sq
rt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*
d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e
^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d
^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 +
792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*
a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2
*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 2
0*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))) - 4*(9*a
^2*c*d^3*e^3 + a^3*d*e^5 + (9*a*c^2*d^2*e^4 + a^2*c*e^6)*x^6 + (9*a*c^2*d^3*e^3
+ a^2*c*d*e^5)*x^4 + (9*a^2*c*d^2*e^4 + a^3*e^6)*x^2)*sqrt(-e/d)*log((e*x^2 + 2*
d*x*sqrt(-e/d) - d)/(e*x^2 + d)) - 4*(c^3*d^6 + a^2*c*d^2*e^4 + 2*a^3*e^6)*x)/(a
^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^7*e +
3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^6 + (a*c^4*d^8 + 3*a^2*c^
3*d^6*e^2 + 3*a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6)*x^4 + (a^2*c^3*d^7*e + 3*a^3*c^2*
d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x^2), -1/16*(8*(c^3*d^4*e^2 - a^2*c*e^6)*
x^5 + 4*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x^3 - 8*(9*a^2*c*d^3*e^3 + a
^3*d*e^5 + (9*a*c^2*d^2*e^4 + a^2*c*e^6)*x^6 + (9*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x
^4 + (9*a^2*c*d^2*e^4 + a^3*e^6)*x^2)*sqrt(e/d)*arctan(e*x/(d*sqrt(e/d))) - (a^2
*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^7*e + 3*
a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^6 + (a*c^4*d^8 + 3*a^2*c^3*
d^6*e^2 + 3*a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6)*x^4 + (a^2*c^3*d^7*e + 3*a^3*c^2*d^
5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x^2)*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3
+ 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 + (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2
+ 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^1
0 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4
- 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 2450
04*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24
+ 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^1
1*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*
d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20
+ 12*a^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5
*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*
e^12))*log((81*c^7*d^10 + 1053*a*c^6*d^8*e^2 + 3602*a^2*c^5*d^6*e^4 - 2958*a^3*c
^4*d^4*e^6 - 23667*a^4*c^3*d^2*e^8 + 2401*a^5*c^2*e^10)*x + (27*a^2*c^7*d^12 + 3
12*a^3*c^6*d^10*e^2 + 843*a^4*c^5*d^8*e^4 - 1592*a^5*c^4*d^6*e^6 - 5967*a^6*c^3*
d^4*e^8 + 4032*a^7*c^2*d^2*e^10 - 343*a^8*c*e^12 + 2*(a^6*c^7*d^15*e + 15*a^7*c^
6*d^13*e^3 + 69*a^8*c^5*d^11*e^5 + 155*a^9*c^4*d^9*e^7 + 195*a^10*c^3*d^7*e^9 +
141*a^11*c^2*d^5*e^11 + 55*a^12*c*d^3*e^13 + 9*a^13*d*e^15)*sqrt(-(81*c^11*d^16
+ 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a
^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^
4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c
^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14
*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16
+ 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)
))*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*
e^7 + (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e
^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 122
4*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^
7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2
*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d
^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10
+ 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220
*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))/(a
^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*
a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))) + (a^2*c^3*d^8 + 3*a^3*c^2*d^6*
e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^7*e + 3*a^2*c^3*d^5*e^3 + 3*a^3*c
^2*d^3*e^5 + a^4*c*d*e^7)*x^6 + (a*c^4*d^8 + 3*a^2*c^3*d^6*e^2 + 3*a^3*c^2*d^4*e
^4 + a^4*c*d^2*e^6)*x^4 + (a^2*c^3*d^7*e + 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 +
a^5*d*e^7)*x^2)*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 -
252*a^3*c^2*d*e^7 + (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20
*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c
^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6
- 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 482
16*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 +
66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12
*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*
d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a
^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3
*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))*log((81*c^7*d^10 +
1053*a*c^6*d^8*e^2 + 3602*a^2*c^5*d^6*e^4 - 2958*a^3*c^4*d^4*e^6 - 23667*a^4*c^
3*d^2*e^8 + 2401*a^5*c^2*e^10)*x - (27*a^2*c^7*d^12 + 312*a^3*c^6*d^10*e^2 + 843
*a^4*c^5*d^8*e^4 - 1592*a^5*c^4*d^6*e^6 - 5967*a^6*c^3*d^4*e^8 + 4032*a^7*c^2*d^
2*e^10 - 343*a^8*c*e^12 + 2*(a^6*c^7*d^15*e + 15*a^7*c^6*d^13*e^3 + 69*a^8*c^5*d
^11*e^5 + 155*a^9*c^4*d^9*e^7 + 195*a^10*c^3*d^7*e^9 + 141*a^11*c^2*d^5*e^11 + 5
5*a^12*c*d^3*e^13 + 9*a^13*d*e^15)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 +
5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^
5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3
*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c
^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12
*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 +
66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))*sqrt((12*c^5*d^7*e + 1
56*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 + (a^3*c^6*d^12 + 6*a
^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 +
6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*
a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6
*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16
)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^
18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12
+ 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a
^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d
^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c
*d^2*e^10 + a^9*e^12))) - (a^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a
^5*d^2*e^6 + (a*c^4*d^7*e + 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)
*x^6 + (a*c^4*d^8 + 3*a^2*c^3*d^6*e^2 + 3*a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6)*x^4 +
(a^2*c^3*d^7*e + 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x^2)*sqrt((12
*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 - (a^3*
c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7
*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^
14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 -
22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 240
1*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 2
20*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13
*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d
^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12
+ 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4
*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))*log((81*c^7*d^10 + 1053*a*c^6*d^8*e^2 + 360
2*a^2*c^5*d^6*e^4 - 2958*a^3*c^4*d^4*e^6 - 23667*a^4*c^3*d^2*e^8 + 2401*a^5*c^2*
e^10)*x + (27*a^2*c^7*d^12 + 312*a^3*c^6*d^10*e^2 + 843*a^4*c^5*d^8*e^4 - 1592*a
^5*c^4*d^6*e^6 - 5967*a^6*c^3*d^4*e^8 + 4032*a^7*c^2*d^2*e^10 - 343*a^8*c*e^12 -
2*(a^6*c^7*d^15*e + 15*a^7*c^6*d^13*e^3 + 69*a^8*c^5*d^11*e^5 + 155*a^9*c^4*d^9
*e^7 + 195*a^10*c^3*d^7*e^9 + 141*a^11*c^2*d^5*e^11 + 55*a^12*c*d^3*e^13 + 9*a^1
3*d*e^15)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4
776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a
^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 1
2*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^
8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10
*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 1
2*a^18*c*d^2*e^22 + a^19*e^24)))*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*a^
2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 - (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*
c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e
^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a
^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^
5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8
*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^1
6*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14
+ 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^1
8*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*
e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))) +
(a^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^7*e
+ 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^6 + (a*c^4*d^8 + 3*a^2
*c^3*d^6*e^2 + 3*a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6)*x^4 + (a^2*c^3*d^7*e + 3*a^3*c
^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x^2)*sqrt((12*c^5*d^7*e + 156*a*c^4*d^
5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c^2*d*e^7 - (a^3*c^6*d^12 + 6*a^4*c^5*d^10
*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^
2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^1
2*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 +
245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12
*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 49
5*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14
*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4
*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 1
5*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 +
a^9*e^12))*log((81*c^7*d^10 + 1053*a*c^6*d^8*e^2 + 3602*a^2*c^5*d^6*e^4 - 2958*
a^3*c^4*d^4*e^6 - 23667*a^4*c^3*d^2*e^8 + 2401*a^5*c^2*e^10)*x - (27*a^2*c^7*d^1
2 + 312*a^3*c^6*d^10*e^2 + 843*a^4*c^5*d^8*e^4 - 1592*a^5*c^4*d^6*e^6 - 5967*a^6
*c^3*d^4*e^8 + 4032*a^7*c^2*d^2*e^10 - 343*a^8*c*e^12 - 2*(a^6*c^7*d^15*e + 15*a
^7*c^6*d^13*e^3 + 69*a^8*c^5*d^11*e^5 + 155*a^9*c^4*d^9*e^7 + 195*a^10*c^3*d^7*e
^9 + 141*a^11*c^2*d^5*e^11 + 55*a^12*c*d^3*e^13 + 9*a^13*d*e^15)*sqrt(-(81*c^11*
d^16 + 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65
130*a^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a
^7*c^4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*
a^9*c^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7
*d^14*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*
e^16 + 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*
e^24)))*sqrt((12*c^5*d^7*e + 156*a*c^4*d^5*e^3 + 404*a^2*c^3*d^3*e^5 - 252*a^3*c
^2*d*e^7 - (a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*
d^6*e^6 + 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12)*sqrt(-(81*c^11*d^16
+ 1224*a*c^10*d^14*e^2 + 5164*a^2*c^9*d^12*e^4 - 4776*a^3*c^8*d^10*e^6 - 65130*a
^4*c^7*d^8*e^8 - 22856*a^5*c^6*d^6*e^10 + 245004*a^6*c^5*d^4*e^12 - 48216*a^7*c^
4*d^2*e^14 + 2401*a^8*c^3*e^16)/(a^7*c^12*d^24 + 12*a^8*c^11*d^22*e^2 + 66*a^9*c
^10*d^20*e^4 + 220*a^10*c^9*d^18*e^6 + 495*a^11*c^8*d^16*e^8 + 792*a^12*c^7*d^14
*e^10 + 924*a^13*c^6*d^12*e^12 + 792*a^14*c^5*d^10*e^14 + 495*a^15*c^4*d^8*e^16
+ 220*a^16*c^3*d^6*e^18 + 66*a^17*c^2*d^4*e^20 + 12*a^18*c*d^2*e^22 + a^19*e^24)
))/(a^3*c^6*d^12 + 6*a^4*c^5*d^10*e^2 + 15*a^5*c^4*d^8*e^4 + 20*a^6*c^3*d^6*e^6
+ 15*a^7*c^2*d^4*e^8 + 6*a^8*c*d^2*e^10 + a^9*e^12))) - 4*(c^3*d^6 + a^2*c*d^2*e
^4 + 2*a^3*e^6)*x)/(a^2*c^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*
e^6 + (a*c^4*d^7*e + 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^6 +
(a*c^4*d^8 + 3*a^2*c^3*d^6*e^2 + 3*a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6)*x^4 + (a^2*c
^3*d^7*e + 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x^2)]

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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(1/(e*x**2+d)**2/(c*x**4+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.285917, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done