3.74 \(\int \frac{a+b x+c x^2}{\sqrt{-1+x} \sqrt{1+x} (d+e x)^3} \, dx\)

Optimal. Leaf size=199 \[ -\frac{\sqrt{x-1} \sqrt{x+1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac{\tanh ^{-1}\left (\frac{\sqrt{x+1} \sqrt{d+e}}{\sqrt{x-1} \sqrt{d-e}}\right ) \left (d^2 (2 a+c)+e^2 (a+2 c)-3 b d e\right )}{(d-e)^{5/2} (d+e)^{5/2}}+\frac{\sqrt{x-1} \sqrt{x+1} \left (-d e^2 (3 a+4 c)+b d^2 e+2 b e^3+c d^3\right )}{2 e \left (d^2-e^2\right )^2 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.652254, antiderivative size = 242, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156 \[ \frac{\left (1-x^2\right ) \left (a e^2-b d e+c d^2\right )}{2 e \sqrt{x-1} \sqrt{x+1} \left (d^2-e^2\right ) (d+e x)^2}-\frac{\sqrt{x^2-1} \tanh ^{-1}\left (\frac{d x+e}{\sqrt{x^2-1} \sqrt{d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{2 \sqrt{x-1} \sqrt{x+1} \left (d^2-e^2\right )^{5/2}}-\frac{\left (1-x^2\right ) \left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right )}{2 e \sqrt{x-1} \sqrt{x+1} \left (d^2-e^2\right )^2 (d+e x)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)/(Sqrt[-1 + x]*Sqrt[1 + x]*(d + e*x)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 84.8412, size = 306, normalized size = 1.54 \[ \frac{2 c \operatorname{atanh}{\left (\frac{\sqrt{d + e} \sqrt{x + 1}}{\sqrt{d - e} \sqrt{x - 1}} \right )}}{e^{2} \sqrt{d - e} \sqrt{d + e}} - \frac{3 d \sqrt{x - 1} \sqrt{x + 1} \left (a e^{2} - b d e + c d^{2}\right )}{2 e \left (d + e x\right ) \left (d^{2} - e^{2}\right )^{2}} + \frac{2 d \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e} \sqrt{x + 1}}{\sqrt{d - e} \sqrt{x - 1}} \right )}}{e^{2} \left (d - e\right )^{\frac{3}{2}} \left (d + e\right )^{\frac{3}{2}}} - \frac{\sqrt{x - 1} \sqrt{x + 1} \left (b e - 2 c d\right )}{e \left (d + e x\right ) \left (d^{2} - e^{2}\right )} - \frac{\sqrt{x - 1} \sqrt{x + 1} \left (a e^{2} - b d e + c d^{2}\right )}{2 e \left (d + e x\right )^{2} \left (d^{2} - e^{2}\right )} + \frac{\left (2 d^{2} + e^{2}\right ) \left (a e^{2} - b d e + c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e} \sqrt{x + 1}}{\sqrt{d - e} \sqrt{x - 1}} \right )}}{e^{2} \left (d - e\right )^{\frac{5}{2}} \left (d + e\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*atanh(sqrt(d + e)*sqrt(x + 1)/(sqrt(d - e)*sqrt(x - 1)))/(e**2*sqrt(d - e)*s
qrt(d + e)) - 3*d*sqrt(x - 1)*sqrt(x + 1)*(a*e**2 - b*d*e + c*d**2)/(2*e*(d + e*
x)*(d**2 - e**2)**2) + 2*d*(b*e - 2*c*d)*atanh(sqrt(d + e)*sqrt(x + 1)/(sqrt(d -
e)*sqrt(x - 1)))/(e**2*(d - e)**(3/2)*(d + e)**(3/2)) - sqrt(x - 1)*sqrt(x + 1)
*(b*e - 2*c*d)/(e*(d + e*x)*(d**2 - e**2)) - sqrt(x - 1)*sqrt(x + 1)*(a*e**2 - b
*d*e + c*d**2)/(2*e*(d + e*x)**2*(d**2 - e**2)) + (2*d**2 + e**2)*(a*e**2 - b*d*
e + c*d**2)*atanh(sqrt(d + e)*sqrt(x + 1)/(sqrt(d - e)*sqrt(x - 1)))/(e**2*(d -
e)**(5/2)*(d + e)**(5/2))

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Mathematica [A]  time = 0.709456, size = 178, normalized size = 0.89 \[ \frac{\frac{2 \tan ^{-1}\left (\frac{\sqrt{\frac{x-1}{x+1}} \sqrt{e-d}}{\sqrt{d+e}}\right ) \left (a \left (2 d^2+e^2\right )-3 b d e+c \left (d^2+2 e^2\right )\right )}{\sqrt{e-d} (d+e)^{5/2}}+\frac{\sqrt{x-1} \sqrt{x+1} \left (a e \left (-4 d^2-3 d e x+e^2\right )+b \left (2 d^3+d^2 e x+d e^2+2 e^3 x\right )+c d \left (d^2 x-3 d e-4 e^2 x\right )\right )}{(d+e)^2 (d+e x)^2}}{2 (d-e)^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x + c*x^2)/(Sqrt[-1 + x]*Sqrt[1 + x]*(d + e*x)^3),x]

[Out]

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

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Maple [B]  time = 0.075, size = 1095, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*a*d^2*e^2*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)-2*b*d^3*e*(x^2-1)^(1/2)*((
d^2-e^2)/e^2)^(1/2)-b*d*e^3*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)+3*c*d^2*e^2*(x^2
-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)+2*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+
d*x+e)/(e*x+d))*x^2*a*d^2*e^2-3*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*
x+e)/(e*x+d))*x^2*b*d*e^3+ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)/(
e*x+d))*x^2*c*d^2*e^2+4*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)/(e*
x+d))*x*a*d^3*e+2*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)/(e*x+d))*
x*a*d*e^3-6*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)/(e*x+d))*x*b*d^
2*e^2+2*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)/(e*x+d))*x*c*d^3*e+
4*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)/(e*x+d))*x*c*d*e^3+3*x*a*
d*e^3*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)+2*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2
)^(1/2)*e+d*x+e)/(e*x+d))*a*d^4+ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*
x+e)/(e*x+d))*c*d^4-2*x*b*e^4*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)-a*e^4*(x^2-1)^
(1/2)*((d^2-e^2)/e^2)^(1/2)+ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)
/(e*x+d))*x^2*a*e^4+2*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)/(e*x+
d))*x^2*c*e^4+ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)/(e*x+d))*a*d^
2*e^2-3*ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)/(e*x+d))*b*d^3*e+2*
ln(-2*(-(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)*e+d*x+e)/(e*x+d))*c*d^2*e^2-x*b*d^2*
e^2*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2)-x*c*d^3*e*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^
(1/2)+4*x*c*d*e^3*(x^2-1)^(1/2)*((d^2-e^2)/e^2)^(1/2))*(1+x)^(1/2)*(-1+x)^(1/2)/
(x^2-1)^(1/2)/(d+e)/(d-e)/(d^2-e^2)/(e*x+d)^2/((d^2-e^2)/e^2)^(1/2)/e

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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((c*x^2 + b*x + a)/((e*x + d)^3*sqrt(x + 1)*sqrt(x - 1)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271378, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/((e*x + d)^3*sqrt(x + 1)*sqrt(x - 1)),x, algorithm="fricas")

[Out]

[1/2*((2*b*d^3*e^2 - (4*a + 3*c)*d^2*e^3 + b*d*e^4 + a*e^5 + 2*(2*c*d^4*e - (2*a
+ 5*c)*d^2*e^3 + 3*b*d*e^4 - a*e^5)*x^2 + (2*c*d^5 + 2*b*d^4*e - (6*a + 7*c)*d^
3*e^2 + 5*b*d^2*e^3 - (3*a + 4*c)*d*e^4 + 2*b*e^5)*x)*sqrt(d^2 - e^2)*sqrt(x + 1
)*sqrt(x - 1) + ((2*a + c)*d^4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4 - 2*((2*a +
c)*d^2*e^4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^4 - 4*((2*a + c)*d^3*e^3 - 3*b*d^2*e^
4 + (a + 2*c)*d*e^5)*x^3 - (2*(2*a + c)*d^4*e^2 - 6*b*d^3*e^3 + 3*c*d^2*e^4 + 3*
b*d*e^5 - (a + 2*c)*e^6)*x^2 + 2*(((2*a + c)*d^2*e^4 - 3*b*d*e^5 + (a + 2*c)*e^6
)*x^3 + 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5)*x^2 + ((2*a + c)*d
^4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4)*x)*sqrt(x + 1)*sqrt(x - 1) + 2*((2*a +
c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5)*x)*log((d^3 - d*e^2 - (d^2*e - e^3
+ (e^2*x + d*e)*sqrt(d^2 - e^2))*sqrt(x + 1)*sqrt(x - 1) + (d^2*e - e^3)*x + (e^
2*x^2 + d*e*x + d^2 - e^2)*sqrt(d^2 - e^2))/(e*x^2 - (e*x + d)*sqrt(x + 1)*sqrt(
x - 1) + d*x)) + (c*d^5 + b*d^4*e - (3*a + 4*c)*d^3*e^2 + 2*b*d^2*e^3 - 2*(2*c*d
^4*e - (2*a + 5*c)*d^2*e^3 + 3*b*d*e^4 - a*e^5)*x^3 - (2*c*d^5 + 2*b*d^4*e - (6*
a + 7*c)*d^3*e^2 + 5*b*d^2*e^3 - (3*a + 4*c)*d*e^4 + 2*b*e^5)*x^2 + 2*(c*d^4*e -
b*d^3*e^2 + (a - c)*d^2*e^3 + b*d*e^4 - a*e^5)*x)*sqrt(d^2 - e^2))/(2*((d^4*e^4
- 2*d^2*e^6 + e^8)*x^3 + 2*(d^5*e^3 - 2*d^3*e^5 + d*e^7)*x^2 + (d^6*e^2 - 2*d^4
*e^4 + d^2*e^6)*x)*sqrt(d^2 - e^2)*sqrt(x + 1)*sqrt(x - 1) + (d^6*e^2 - 2*d^4*e^
4 + d^2*e^6 - 2*(d^4*e^4 - 2*d^2*e^6 + e^8)*x^4 - 4*(d^5*e^3 - 2*d^3*e^5 + d*e^7
)*x^3 - (2*d^6*e^2 - 5*d^4*e^4 + 4*d^2*e^6 - e^8)*x^2 + 2*(d^5*e^3 - 2*d^3*e^5 +
d*e^7)*x)*sqrt(d^2 - e^2)), 1/2*((2*b*d^3*e^2 - (4*a + 3*c)*d^2*e^3 + b*d*e^4 +
a*e^5 + 2*(2*c*d^4*e - (2*a + 5*c)*d^2*e^3 + 3*b*d*e^4 - a*e^5)*x^2 + (2*c*d^5
+ 2*b*d^4*e - (6*a + 7*c)*d^3*e^2 + 5*b*d^2*e^3 - (3*a + 4*c)*d*e^4 + 2*b*e^5)*x
)*sqrt(-d^2 + e^2)*sqrt(x + 1)*sqrt(x - 1) + 2*((2*a + c)*d^4*e^2 - 3*b*d^3*e^3
+ (a + 2*c)*d^2*e^4 - 2*((2*a + c)*d^2*e^4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^4 - 4*
((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5)*x^3 - (2*(2*a + c)*d^4*e^2 -
6*b*d^3*e^3 + 3*c*d^2*e^4 + 3*b*d*e^5 - (a + 2*c)*e^6)*x^2 + 2*(((2*a + c)*d^2*
e^4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^3 + 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a +
2*c)*d*e^5)*x^2 + ((2*a + c)*d^4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4)*x)*sqrt
(x + 1)*sqrt(x - 1) + 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5)*x)*a
rctan(-(sqrt(-d^2 + e^2)*e*sqrt(x + 1)*sqrt(x - 1) - sqrt(-d^2 + e^2)*(e*x + d))
/(d^2 - e^2)) + (c*d^5 + b*d^4*e - (3*a + 4*c)*d^3*e^2 + 2*b*d^2*e^3 - 2*(2*c*d^
4*e - (2*a + 5*c)*d^2*e^3 + 3*b*d*e^4 - a*e^5)*x^3 - (2*c*d^5 + 2*b*d^4*e - (6*a
+ 7*c)*d^3*e^2 + 5*b*d^2*e^3 - (3*a + 4*c)*d*e^4 + 2*b*e^5)*x^2 + 2*(c*d^4*e -
b*d^3*e^2 + (a - c)*d^2*e^3 + b*d*e^4 - a*e^5)*x)*sqrt(-d^2 + e^2))/(2*((d^4*e^4
- 2*d^2*e^6 + e^8)*x^3 + 2*(d^5*e^3 - 2*d^3*e^5 + d*e^7)*x^2 + (d^6*e^2 - 2*d^4
*e^4 + d^2*e^6)*x)*sqrt(-d^2 + e^2)*sqrt(x + 1)*sqrt(x - 1) + (d^6*e^2 - 2*d^4*e
^4 + d^2*e^6 - 2*(d^4*e^4 - 2*d^2*e^6 + e^8)*x^4 - 4*(d^5*e^3 - 2*d^3*e^5 + d*e^
7)*x^3 - (2*d^6*e^2 - 5*d^4*e^4 + 4*d^2*e^6 - e^8)*x^2 + 2*(d^5*e^3 - 2*d^3*e^5
+ d*e^7)*x)*sqrt(-d^2 + e^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((c*x**2+b*x+a)/(e*x+d)**3/(-1+x)**(1/2)/(1+x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.409886, size = 817, normalized size = 4.11 \[ -\frac{{\left (2 \, a d^{2} + c d^{2} - 3 \, b d e + a e^{2} + 2 \, c e^{2}\right )} \arctan \left (\frac{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e + 2 \, d}{2 \, \sqrt{-d^{2} + e^{2}}}\right )}{{\left (d^{4} - 2 \, d^{2} e^{2} + e^{4}\right )} \sqrt{-d^{2} + e^{2}}} + \frac{2 \,{\left (2 \, c d^{4}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{6} e + 4 \, c d^{5}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} - 2 \, a d^{2}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{6} e^{3} - 5 \, c d^{2}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{6} e^{3} + 4 \, b d^{4}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e + 3 \, b d{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{6} e^{4} - 12 \, a d^{3}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{2} - 14 \, c d^{3}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{2} - a{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{6} e^{5} + 10 \, b d^{2}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{3} + 8 \, c d^{4}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e - 6 \, a d{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{4} - 8 \, c d{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{4} + 16 \, b d^{3}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e^{2} + 4 \, b{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e^{5} - 40 \, a d^{2}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e^{3} - 44 \, c d^{2}{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e^{3} + 20 \, b d{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e^{4} + 8 \, c d^{3} e^{2} + 4 \, a{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} e^{5} + 8 \, b d^{2} e^{3} - 24 \, a d e^{4} - 32 \, c d e^{4} + 16 \, b e^{5}\right )}}{{\left (d^{4} e^{2} - 2 \, d^{2} e^{4} + e^{6}\right )}{\left ({\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{4} e + 4 \, d{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} + 4 \, e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/((e*x + d)^3*sqrt(x + 1)*sqrt(x - 1)),x, algorithm="giac")

[Out]

-(2*a*d^2 + c*d^2 - 3*b*d*e + a*e^2 + 2*c*e^2)*arctan(1/2*((sqrt(x + 1) - sqrt(x
- 1))^2*e + 2*d)/sqrt(-d^2 + e^2))/((d^4 - 2*d^2*e^2 + e^4)*sqrt(-d^2 + e^2)) +
2*(2*c*d^4*(sqrt(x + 1) - sqrt(x - 1))^6*e + 4*c*d^5*(sqrt(x + 1) - sqrt(x - 1)
)^4 - 2*a*d^2*(sqrt(x + 1) - sqrt(x - 1))^6*e^3 - 5*c*d^2*(sqrt(x + 1) - sqrt(x
- 1))^6*e^3 + 4*b*d^4*(sqrt(x + 1) - sqrt(x - 1))^4*e + 3*b*d*(sqrt(x + 1) - sqr
t(x - 1))^6*e^4 - 12*a*d^3*(sqrt(x + 1) - sqrt(x - 1))^4*e^2 - 14*c*d^3*(sqrt(x
+ 1) - sqrt(x - 1))^4*e^2 - a*(sqrt(x + 1) - sqrt(x - 1))^6*e^5 + 10*b*d^2*(sqrt
(x + 1) - sqrt(x - 1))^4*e^3 + 8*c*d^4*(sqrt(x + 1) - sqrt(x - 1))^2*e - 6*a*d*(
sqrt(x + 1) - sqrt(x - 1))^4*e^4 - 8*c*d*(sqrt(x + 1) - sqrt(x - 1))^4*e^4 + 16*
b*d^3*(sqrt(x + 1) - sqrt(x - 1))^2*e^2 + 4*b*(sqrt(x + 1) - sqrt(x - 1))^4*e^5
- 40*a*d^2*(sqrt(x + 1) - sqrt(x - 1))^2*e^3 - 44*c*d^2*(sqrt(x + 1) - sqrt(x -
1))^2*e^3 + 20*b*d*(sqrt(x + 1) - sqrt(x - 1))^2*e^4 + 8*c*d^3*e^2 + 4*a*(sqrt(x
+ 1) - sqrt(x - 1))^2*e^5 + 8*b*d^2*e^3 - 24*a*d*e^4 - 32*c*d*e^4 + 16*b*e^5)/(
(d^4*e^2 - 2*d^2*e^4 + e^6)*((sqrt(x + 1) - sqrt(x - 1))^4*e + 4*d*(sqrt(x + 1)
- sqrt(x - 1))^2 + 4*e)^2)