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

Optimal. Leaf size=751 \[ \frac{c^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (2 a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{c^{5/4} \left (\sqrt{a} e+3 \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}-\frac{c^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (2 a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{c^{5/4} \left (\sqrt{a} e+3 \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}+\frac{c^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{c^{5/4} \left (3 \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}-\frac{c^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{c^{5/4} \left (3 \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}-\frac{c^2 x \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{e}{a^2 d^2 x}-\frac{1}{3 a^2 d x^3}+\frac{e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (a e^2+c d^2\right )^2} \]

[Out]

-1/(3*a^2*d*x^3) + e/(a^2*d^2*x) - (c^2*x*(d - e*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a
+ c*x^4)) + (e^(11/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*(c*d^2 + a*e^2)^2)
+ (c^(5/4)*(3*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8
*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) + (c^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + 2
*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*(c*d^2 + a*
e^2)^2) - (c^(5/4)*(3*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1
/4)])/(8*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c
*d^2 + 2*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*(c*
d^2 + a*e^2)^2) + (c^(5/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/
4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) + (c^(5/4)*(S
qrt[c]*d + Sqrt[a]*e)*(c*d^2 + 2*a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x
+ Sqrt[c]*x^2])/(4*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) - (c^(5/4)*(3*Sqrt[c]*d +
Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*
a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 + 2*a*e^2)*L
og[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(11/4)*(c*d^
2 + a*e^2)^2)

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Rubi [A]  time = 1.32395, antiderivative size = 751, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ \frac{c^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (2 a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{c^{5/4} \left (\sqrt{a} e+3 \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}-\frac{c^{5/4} \left (\sqrt{a} e+\sqrt{c} d\right ) \left (2 a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{c^{5/4} \left (\sqrt{a} e+3 \sqrt{c} d\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}+\frac{c^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{c^{5/4} \left (3 \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}-\frac{c^{5/4} \left (\sqrt{c} d-\sqrt{a} e\right ) \left (2 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{c^{5/4} \left (3 \sqrt{c} d-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{11/4} \left (a e^2+c d^2\right )}-\frac{c^2 x \left (d-e x^2\right )}{4 a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{e}{a^2 d^2 x}-\frac{1}{3 a^2 d x^3}+\frac{e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(3*a^2*d*x^3) + e/(a^2*d^2*x) - (c^2*x*(d - e*x^2))/(4*a^2*(c*d^2 + a*e^2)*(a
+ c*x^4)) + (e^(11/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*(c*d^2 + a*e^2)^2)
+ (c^(5/4)*(3*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8
*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) + (c^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + 2
*a*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*(c*d^2 + a*
e^2)^2) - (c^(5/4)*(3*Sqrt[c]*d - Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1
/4)])/(8*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c
*d^2 + 2*a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(11/4)*(c*
d^2 + a*e^2)^2) + (c^(5/4)*(3*Sqrt[c]*d + Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/
4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)) + (c^(5/4)*(S
qrt[c]*d + Sqrt[a]*e)*(c*d^2 + 2*a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x
+ Sqrt[c]*x^2])/(4*Sqrt[2]*a^(11/4)*(c*d^2 + a*e^2)^2) - (c^(5/4)*(3*Sqrt[c]*d +
Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*
a^(11/4)*(c*d^2 + a*e^2)) - (c^(5/4)*(Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 + 2*a*e^2)*L
og[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(11/4)*(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/x**4/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.714258, size = 513, normalized size = 0.68 \[ \frac{1}{96} \left (\frac{3 \sqrt{2} c^{5/4} \left (9 a^{3/2} e^3+5 \sqrt{a} c d^2 e+11 a \sqrt{c} d e^2+7 c^{3/2} d^3\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{3 \sqrt{2} c^{5/4} \left (9 a^{3/2} e^3+5 \sqrt{a} c d^2 e+11 a \sqrt{c} d e^2+7 c^{3/2} d^3\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{6 \sqrt{2} c^{5/4} \left (-9 a^{3/2} e^3-5 \sqrt{a} c d^2 e+11 a \sqrt{c} d e^2+7 c^{3/2} d^3\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4} \left (a e^2+c d^2\right )^2}+\frac{6 \sqrt{2} c^{5/4} \left (9 a^{3/2} e^3+5 \sqrt{a} c d^2 e-11 a \sqrt{c} d e^2-7 c^{3/2} d^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{11/4} \left (a e^2+c d^2\right )^2}-\frac{24 c^2 x \left (d-e x^2\right )}{a^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{96 e}{a^2 d^2 x}-\frac{32}{a^2 d x^3}+\frac{96 e^{11/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d^{5/2} \left (a e^2+c d^2\right )^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-32/(a^2*d*x^3) + (96*e)/(a^2*d^2*x) - (24*c^2*x*(d - e*x^2))/(a^2*(c*d^2 + a*e
^2)*(a + c*x^4)) + (96*e^(11/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*(c*d^2 + a
*e^2)^2) + (6*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 - 5*Sqrt[a]*c*d^2*e + 11*a*Sqrt[c]*
d*e^2 - 9*a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(11/4)*(c*d^2
+ a*e^2)^2) + (6*Sqrt[2]*c^(5/4)*(-7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e - 11*a*Sqr
t[c]*d*e^2 + 9*a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(a^(11/4)*(
c*d^2 + a*e^2)^2) + (3*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 + 5*Sqrt[a]*c*d^2*e + 11*a
*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c
]*x^2])/(a^(11/4)*(c*d^2 + a*e^2)^2) - (3*Sqrt[2]*c^(5/4)*(7*c^(3/2)*d^3 + 5*Sqr
t[a]*c*d^2*e + 11*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)
*c^(1/4)*x + Sqrt[c]*x^2])/(a^(11/4)*(c*d^2 + a*e^2)^2))/96

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Maple [A]  time = 0.03, size = 932, normalized size = 1.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/a^2/d/x^3+e/a^2/d^2/x+1/4*c^2/(a*e^2+c*d^2)^2/a/(c*x^4+a)*x^3*e^3+1/4*c^3/(
a*e^2+c*d^2)^2/a^2/(c*x^4+a)*x^3*d^2*e-1/4*c^2/(a*e^2+c*d^2)^2/a/(c*x^4+a)*e^2*d
*x-1/4*c^3/(a*e^2+c*d^2)^2/a^2/(c*x^4+a)*d^3*x-11/16*c^2/(a*e^2+c*d^2)^2/a^2*(1/
c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d*e^2-7/16*c^3/(a*e^2+c*d^2
)^2/a^3*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^3-11/16*c^2/(a
*e^2+c*d^2)^2/a^2*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d*e^2-
7/16*c^3/(a*e^2+c*d^2)^2/a^3*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*
x-1)*d^3-11/32*c^2/(a*e^2+c*d^2)^2/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*e^2-7
/32*c^3/(a*e^2+c*d^2)^2/a^3*(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+9/32*c/(a*e^2+
c*d^2)^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)))*e^3+5/32*c^2/(a*e^2+c*d^2)^2/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^2*e+9/16*c/(a*e^2+c*d^2)^2/a/(1/c*a)^(1/4)*2^(
1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*e^3+5/16*c^2/(a*e^2+c*d^2)^2/a^2/(1/c*a)^
(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^2*e+9/16*c/(a*e^2+c*d^2)^2/a/(
1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*e^3+5/16*c^2/(a*e^2+c*d^2
)^2/a^2/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^2*e+1/d^2*e^6/
(a*e^2+c*d^2)^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))

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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)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 106.225, 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)*x^4),x, algorithm="fricas")

[Out]

[-1/48*(16*a*c^2*d^5 + 32*a^2*c*d^3*e^2 + 16*a^3*d*e^4 - 12*(5*c^3*d^4*e + 9*a*c
^2*d^2*e^3 + 4*a^2*c*e^5)*x^6 + 4*(7*c^3*d^5 + 11*a*c^2*d^3*e^2 + 4*a^2*c*d*e^4)
*x^4 - 48*(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)*x^2 - 3*((a^2*c^3*d^6 + 2*a^
3*c^2*d^4*e^2 + a^4*c*d^2*e^4)*x^7 + (a^3*c^2*d^6 + 2*a^4*c*d^4*e^2 + a^5*d^2*e^
4)*x^3)*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 + (a^5*c^4*d^
8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(24
01*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^
6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8
*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a
^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^1
4 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*
d^2*e^6 + a^9*e^8))*log(-(2401*c^8*d^8 + 10290*a*c^7*d^6*e^2 + 11968*a^2*c^6*d^4
*e^4 - 1458*a^3*c^5*d^2*e^6 - 6561*a^4*c^4*e^8)*x + (343*a^3*c^7*d^9 + 1442*a^4*
c^6*d^7*e^2 + 1636*a^5*c^5*d^5*e^4 - 226*a^6*c^4*d^3*e^6 - 891*a^7*c^3*d*e^8 + (
5*a^9*c^5*d^10*e + 29*a^10*c^4*d^8*e^3 + 66*a^11*c^3*d^6*e^5 + 74*a^12*c^2*d^4*e
^7 + 41*a^13*c*d^2*e^9 + 9*a^14*e^11)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*
e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*
a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*
a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6
*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))*sqrt((70*c^5*d^5
*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 + (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 +
6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c
^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^
8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*
e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^1
6*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^
4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))) + 3
*((a^2*c^3*d^6 + 2*a^3*c^2*d^4*e^2 + a^4*c*d^2*e^4)*x^7 + (a^3*c^2*d^6 + 2*a^4*c
*d^4*e^2 + a^5*d^2*e^4)*x^3)*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^
3*d*e^5 + (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6
+ a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^
4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^
6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^1
4*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^
12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c
^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))*log(-(2401*c^8*d^8 + 10290*a*c^7*d^6*e^
2 + 11968*a^2*c^6*d^4*e^4 - 1458*a^3*c^5*d^2*e^6 - 6561*a^4*c^4*e^8)*x - (343*a^
3*c^7*d^9 + 1442*a^4*c^6*d^7*e^2 + 1636*a^5*c^5*d^5*e^4 - 226*a^6*c^4*d^3*e^6 -
891*a^7*c^3*d*e^8 + (5*a^9*c^5*d^10*e + 29*a^10*c^4*d^8*e^3 + 66*a^11*c^3*d^6*e^
5 + 74*a^12*c^2*d^4*e^7 + 41*a^13*c*d^2*e^9 + 9*a^14*e^11)*sqrt(-(2401*c^11*d^12
+ 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^
4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^
12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*
e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^1
6)))*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 + (a^5*c^4*d^8 +
4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*
c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 -
19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^
16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15
*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 +
a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2
*e^6 + a^9*e^8))) - 3*((a^2*c^3*d^6 + 2*a^3*c^2*d^4*e^2 + a^4*c*d^2*e^4)*x^7 + (
a^3*c^2*d^6 + 2*a^4*c*d^4*e^2 + a^5*d^2*e^4)*x^3)*sqrt((70*c^5*d^5*e + 236*a*c^4
*d^3*e^3 + 198*a^2*c^3*d*e^5 - (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*
e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 +
19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c
^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*
c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10
+ 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*
c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))*log(-(2401*c^8*d^8
+ 10290*a*c^7*d^6*e^2 + 11968*a^2*c^6*d^4*e^4 - 1458*a^3*c^5*d^2*e^6 - 6561*a^4
*c^4*e^8)*x + (343*a^3*c^7*d^9 + 1442*a^4*c^6*d^7*e^2 + 1636*a^5*c^5*d^5*e^4 - 2
26*a^6*c^4*d^3*e^6 - 891*a^7*c^3*d*e^8 - (5*a^9*c^5*d^10*e + 29*a^10*c^4*d^8*e^3
+ 66*a^11*c^3*d^6*e^5 + 74*a^12*c^2*d^4*e^7 + 41*a^13*c*d^2*e^9 + 9*a^14*e^11)*
sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c
^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/
(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e
^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*
c*d^2*e^14 + a^19*e^16)))*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d
*e^5 - (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 +
a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 +
60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c
^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c
^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12
+ 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*
d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))) + 3*((a^2*c^3*d^6 + 2*a^3*c^2*d^4*e^2 + a
^4*c*d^2*e^4)*x^7 + (a^3*c^2*d^6 + 2*a^4*c*d^4*e^2 + a^5*d^2*e^4)*x^3)*sqrt((70*
c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 - (a^5*c^4*d^8 + 4*a^6*c^3*d^6
*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12
642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7
*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^
7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 +
56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/
(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8
))*log(-(2401*c^8*d^8 + 10290*a*c^7*d^6*e^2 + 11968*a^2*c^6*d^4*e^4 - 1458*a^3*c
^5*d^2*e^6 - 6561*a^4*c^4*e^8)*x - (343*a^3*c^7*d^9 + 1442*a^4*c^6*d^7*e^2 + 163
6*a^5*c^5*d^5*e^4 - 226*a^6*c^4*d^3*e^6 - 891*a^7*c^3*d*e^8 - (5*a^9*c^5*d^10*e
+ 29*a^10*c^4*d^8*e^3 + 66*a^11*c^3*d^6*e^5 + 74*a^12*c^2*d^4*e^7 + 41*a^13*c*d^
2*e^9 + 9*a^14*e^11)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c
^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10
+ 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4
+ 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c
^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))*sqrt((70*c^5*d^5*e + 236*a*c^4*d^
3*e^3 + 198*a^2*c^3*d*e^5 - (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4
+ 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19
679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*
d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6
*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 +
28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3
*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))) - 24*(a^2*c*e^5*x^7
+ a^3*e^5*x^3)*sqrt(-e/d)*log((e*x^2 + 2*d*x*sqrt(-e/d) - d)/(e*x^2 + d)))/((a^2
*c^3*d^6 + 2*a^3*c^2*d^4*e^2 + a^4*c*d^2*e^4)*x^7 + (a^3*c^2*d^6 + 2*a^4*c*d^4*e
^2 + a^5*d^2*e^4)*x^3), -1/48*(16*a*c^2*d^5 + 32*a^2*c*d^3*e^2 + 16*a^3*d*e^4 -
12*(5*c^3*d^4*e + 9*a*c^2*d^2*e^3 + 4*a^2*c*e^5)*x^6 + 4*(7*c^3*d^5 + 11*a*c^2*d
^3*e^2 + 4*a^2*c*d*e^4)*x^4 - 48*(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)*x^2 -
48*(a^2*c*e^5*x^7 + a^3*e^5*x^3)*sqrt(e/d)*arctan(e*x/(d*sqrt(e/d))) - 3*((a^2*
c^3*d^6 + 2*a^3*c^2*d^4*e^2 + a^4*c*d^2*e^4)*x^7 + (a^3*c^2*d^6 + 2*a^4*c*d^4*e^
2 + a^5*d^2*e^4)*x^3)*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5
+ (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*
e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*
a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e
^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d
^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*
a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*
e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))*log(-(2401*c^8*d^8 + 10290*a*c^7*d^6*e^2 + 119
68*a^2*c^6*d^4*e^4 - 1458*a^3*c^5*d^2*e^6 - 6561*a^4*c^4*e^8)*x + (343*a^3*c^7*d
^9 + 1442*a^4*c^6*d^7*e^2 + 1636*a^5*c^5*d^5*e^4 - 226*a^6*c^4*d^3*e^6 - 891*a^7
*c^3*d*e^8 + (5*a^9*c^5*d^10*e + 29*a^10*c^4*d^8*e^3 + 66*a^11*c^3*d^6*e^5 + 74*
a^12*c^2*d^4*e^7 + 41*a^13*c*d^2*e^9 + 9*a^14*e^11)*sqrt(-(2401*c^11*d^12 + 1264
2*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d
^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*
d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 5
6*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))*sq
rt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 + (a^5*c^4*d^8 + 4*a^6*
c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^
12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*
a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*
a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^
8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e
^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 +
a^9*e^8))) + 3*((a^2*c^3*d^6 + 2*a^3*c^2*d^4*e^2 + a^4*c*d^2*e^4)*x^7 + (a^3*c^2
*d^6 + 2*a^4*c*d^4*e^2 + a^5*d^2*e^4)*x^3)*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^
3 + 198*a^2*c^3*d*e^5 + (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4
*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*
a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*
e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^1
2*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a
^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6
*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))*log(-(2401*c^8*d^8 + 1029
0*a*c^7*d^6*e^2 + 11968*a^2*c^6*d^4*e^4 - 1458*a^3*c^5*d^2*e^6 - 6561*a^4*c^4*e^
8)*x - (343*a^3*c^7*d^9 + 1442*a^4*c^6*d^7*e^2 + 1636*a^5*c^5*d^5*e^4 - 226*a^6*
c^4*d^3*e^6 - 891*a^7*c^3*d*e^8 + (5*a^9*c^5*d^10*e + 29*a^10*c^4*d^8*e^3 + 66*a
^11*c^3*d^6*e^5 + 74*a^12*c^2*d^4*e^7 + 41*a^13*c*d^2*e^9 + 9*a^14*e^11)*sqrt(-(
2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*
e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c
^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70
*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e
^14 + a^19*e^16)))*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 +
(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8
)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3
*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12
)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10
*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^1
8*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4
+ 4*a^8*c*d^2*e^6 + a^9*e^8))) - 3*((a^2*c^3*d^6 + 2*a^3*c^2*d^4*e^2 + a^4*c*d^
2*e^4)*x^7 + (a^3*c^2*d^6 + 2*a^4*c*d^4*e^2 + a^5*d^2*e^4)*x^3)*sqrt((70*c^5*d^5
*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 - (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 +
6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c
^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^
8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*
e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^1
6*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^
4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))*log(
-(2401*c^8*d^8 + 10290*a*c^7*d^6*e^2 + 11968*a^2*c^6*d^4*e^4 - 1458*a^3*c^5*d^2*
e^6 - 6561*a^4*c^4*e^8)*x + (343*a^3*c^7*d^9 + 1442*a^4*c^6*d^7*e^2 + 1636*a^5*c
^5*d^5*e^4 - 226*a^6*c^4*d^3*e^6 - 891*a^7*c^3*d*e^8 - (5*a^9*c^5*d^10*e + 29*a^
10*c^4*d^8*e^3 + 66*a^11*c^3*d^6*e^5 + 74*a^12*c^2*d^4*e^7 + 41*a^13*c*d^2*e^9 +
9*a^14*e^11)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*
e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*
a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a
^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*
e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 +
198*a^2*c^3*d*e^5 - (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^
8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2
*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^1
0 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e
^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17
*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^
2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))) + 3*((a^2*c^3*d^6 + 2*a^3*c
^2*d^4*e^2 + a^4*c*d^2*e^4)*x^7 + (a^3*c^2*d^6 + 2*a^4*c*d^4*e^2 + a^5*d^2*e^4)*
x^3)*sqrt((70*c^5*d^5*e + 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 - (a^5*c^4*d^8 +
4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*
c^11*d^12 + 12642*a*c^10*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 -
19937*a^4*c^7*d^4*e^8 - 5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^
16 + 8*a^12*c^7*d^14*e^2 + 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15
*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 +
a^19*e^16)))/(a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2
*e^6 + a^9*e^8))*log(-(2401*c^8*d^8 + 10290*a*c^7*d^6*e^2 + 11968*a^2*c^6*d^4*e^
4 - 1458*a^3*c^5*d^2*e^6 - 6561*a^4*c^4*e^8)*x - (343*a^3*c^7*d^9 + 1442*a^4*c^6
*d^7*e^2 + 1636*a^5*c^5*d^5*e^4 - 226*a^6*c^4*d^3*e^6 - 891*a^7*c^3*d*e^8 - (5*a
^9*c^5*d^10*e + 29*a^10*c^4*d^8*e^3 + 66*a^11*c^3*d^6*e^5 + 74*a^12*c^2*d^4*e^7
+ 41*a^13*c*d^2*e^9 + 9*a^14*e^11)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10*d^10*e^2
+ 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 - 5022*a^5
*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2 + 28*a^1
3*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c^3*d^6*e^
10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))*sqrt((70*c^5*d^5*e
+ 236*a*c^4*d^3*e^3 + 198*a^2*c^3*d*e^5 - (a^5*c^4*d^8 + 4*a^6*c^3*d^6*e^2 + 6*a
^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8)*sqrt(-(2401*c^11*d^12 + 12642*a*c^10
*d^10*e^2 + 19679*a^2*c^9*d^8*e^4 + 60*a^3*c^8*d^6*e^6 - 19937*a^4*c^7*d^4*e^8 -
5022*a^5*c^6*d^2*e^10 + 6561*a^6*c^5*e^12)/(a^11*c^8*d^16 + 8*a^12*c^7*d^14*e^2
+ 28*a^13*c^6*d^12*e^4 + 56*a^14*c^5*d^10*e^6 + 70*a^15*c^4*d^8*e^8 + 56*a^16*c
^3*d^6*e^10 + 28*a^17*c^2*d^4*e^12 + 8*a^18*c*d^2*e^14 + a^19*e^16)))/(a^5*c^4*d
^8 + 4*a^6*c^3*d^6*e^2 + 6*a^7*c^2*d^4*e^4 + 4*a^8*c*d^2*e^6 + a^9*e^8))))/((a^2
*c^3*d^6 + 2*a^3*c^2*d^4*e^2 + a^4*c*d^2*e^4)*x^7 + (a^3*c^2*d^6 + 2*a^4*c*d^4*e
^2 + a^5*d^2*e^4)*x^3)]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.28641, size = 848, normalized size = 1.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2*d*e^2 - 5*(a*c^3)^(3/4)*c
*d^2*e - 9*(a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(
a/c)^(1/4))/(sqrt(2)*a^3*c^3*d^4 + 2*sqrt(2)*a^4*c^2*d^2*e^2 + sqrt(2)*a^5*c*e^4
) - 1/8*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2*d*e^2 - 5*(a*c^3)^(3/4
)*c*d^2*e - 9*(a*c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4)
)/(a/c)^(1/4))/(sqrt(2)*a^3*c^3*d^4 + 2*sqrt(2)*a^4*c^2*d^2*e^2 + sqrt(2)*a^5*c*
e^4) - 1/16*(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2*d*e^2 + 5*(a*c^3)^
(3/4)*c*d^2*e + 9*(a*c^3)^(3/4)*a*e^3)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c
))/(sqrt(2)*a^3*c^3*d^4 + 2*sqrt(2)*a^4*c^2*d^2*e^2 + sqrt(2)*a^5*c*e^4) + 1/16*
(7*(a*c^3)^(1/4)*c^3*d^3 + 11*(a*c^3)^(1/4)*a*c^2*d*e^2 + 5*(a*c^3)^(3/4)*c*d^2*
e + 9*(a*c^3)^(3/4)*a*e^3)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*
a^3*c^3*d^4 + 2*sqrt(2)*a^4*c^2*d^2*e^2 + sqrt(2)*a^5*c*e^4) + arctan(x*e^(1/2)/
sqrt(d))*e^(11/2)/((c^2*d^6 + 2*a*c*d^4*e^2 + a^2*d^2*e^4)*sqrt(d)) + 1/4*(c^2*x
^3*e - c^2*d*x)/((a^2*c*d^2 + a^3*e^2)*(c*x^4 + a)) + 1/3*(3*x^2*e - d)/(a^2*d^2
*x^3)