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

Optimal. Leaf size=195 \[ -\frac{\sqrt{x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac{\sqrt{x^2-1} \left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right )}{2 e \left (d^2-e^2\right )^2 (d+e x)}-\frac{\tanh ^{-1}\left (\frac{d x+e}{\sqrt{x^2-1} \sqrt{d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{2 \left (d^2-e^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.206653, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1651, 807, 725, 206} \[ -\frac{\sqrt{x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac{\sqrt{x^2-1} \left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right )}{2 e \left (d^2-e^2\right )^2 (d+e x)}-\frac{\tanh ^{-1}\left (\frac{d x+e}{\sqrt{x^2-1} \sqrt{d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{2 \left (d^2-e^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{a+b x+c x^2}{(d+e x)^3 \sqrt{-1+x^2}} \, dx &=-\frac{\left (c d^2-b d e+a e^2\right ) \sqrt{-1+x^2}}{2 e \left (d^2-e^2\right ) (d+e x)^2}-\frac{\int \frac{-2 (a d+c d-b e)-\left (b d+\frac{c d^2}{e}-a e-2 c e\right ) x}{(d+e x)^2 \sqrt{-1+x^2}} \, dx}{2 \left (d^2-e^2\right )}\\ &=-\frac{\left (c d^2-b d e+a e^2\right ) \sqrt{-1+x^2}}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac{\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 ) \sqrt{-1+x^2}}{2 e \left (d^2-e^2\right )^2 (d+e x)}-\frac{\left (3 b d e-a \left (2 d^2+e^2\right )-c \left (d^2+2 e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{-1+x^2}} \, dx}{2 \left (d^2-e^2\right )^2}\\ &=-\frac{\left (c d^2-b d e+a e^2\right ) \sqrt{-1+x^2}}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac{\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 ) \sqrt{-1+x^2}}{2 e \left (d^2-e^2\right )^2 (d+e x)}+\frac{\left (3 b d e-a \left (2 d^2+e^2\right )-c \left (d^2+2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{d^2-e^2-x^2} \, dx,x,\frac{-e-d x}{\sqrt{-1+x^2}}\right )}{2 \left (d^2-e^2\right )^2}\\ &=-\frac{\left (c d^2-b d e+a e^2\right ) \sqrt{-1+x^2}}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac{\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 ) \sqrt{-1+x^2}}{2 e \left (d^2-e^2\right )^2 (d+e x)}-\frac{\left (3 b d e-a \left (2 d^2+e^2\right )-c \left (d^2+2 e^2\right )\right ) \tanh ^{-1}\left (\frac{e+d x}{\sqrt{d^2-e^2} \sqrt{-1+x^2}}\right )}{2 \left (d^2-e^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.323999, size = 240, normalized size = 1.23 \[ \frac{1}{2} \left (\frac{\sqrt{x^2-1} \left (a e \left (-4 d^2-3 d e x+e^2\right )+b \left (d^2 e x+2 d^3+d e^2+2 e^3 x\right )+c d \left (d^2 x-3 d e-4 e^2 x\right )\right )}{\left (d^2-e^2\right )^2 (d+e x)^2}-\frac{\log \left (-\sqrt{x^2-1} \sqrt{d^2-e^2}+d x+e\right ) \left (a \left (2 d^2+e^2\right )-3 b d e+c \left (d^2+2 e^2\right )\right )}{(d-e)^2 (d+e)^2 \sqrt{d^2-e^2}}+\frac{\log (d+e x) \left (a \left (2 d^2+e^2\right )-3 b d e+c \left (d^2+2 e^2\right )\right )}{(d-e)^2 (d+e)^2 \sqrt{d^2-e^2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.028, size = 1407, normalized size = 7.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/e^3/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e
)+(d^2-e^2)/e^2)^(1/2))/(x+d/e))-1/2/e/(d^2-e^2)/(x+d/e)^2*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*a+1/2
/e^2/(d^2-e^2)/(x+d/e)^2*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*b*d-1/2/e^3/(d^2-e^2)/(x+d/e)^2*((x+d/e
)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*c*d^2-3/2*d/(d^2-e^2)^2/(x+d/e)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)
^(1/2)*a+3/2/e*d^2/(d^2-e^2)^2/(x+d/e)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*b-3/2/e^2*d^3/(d^2-e^2)^2
/(x+d/e)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*c-3/2/e*d^2/(d^2-e^2)^2/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^
2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2))/(x+d/e))*a+3/2
/e^2*d^3/(d^2-e^2)^2/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^
2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2))/(x+d/e))*b-3/2/e^3*d^4/(d^2-e^2)^2/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)
/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2))/(x+d/e))*c+1/2/e/(d^
2-e^2)/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e
)+(d^2-e^2)/e^2)^(1/2))/(x+d/e))*a-3/2/e^2/(d^2-e^2)/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2
*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2))/(x+d/e))*b*d+5/2/e^3/(d^2-e^2)/((d^2-e^2
)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2*d/e*(x+d/e)+2*((d^2-e^2)/e^2)^(1/2)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)
^(1/2))/(x+d/e))*c*d^2-1/e/(d^2-e^2)/(x+d/e)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*b+2/e^2/(d^2-e^2)/(
x+d/e)*((x+d/e)^2-2*d/e*(x+d/e)+(d^2-e^2)/e^2)^(1/2)*c*d

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.99121, size = 2431, normalized size = 12.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(c*d^7 + b*d^6*e - (3*a + 5*c)*d^5*e^2 + b*d^4*e^3 + (3*a + 4*c)*d^3*e^4 - 2*b*d^2*e^5 + (c*d^5*e^2 + b*d
^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x^2 + ((2*a + c)*d^4*e^2 - 3*b*d^3*e^3
+ (a + 2*c)*d^2*e^4 + ((2*a + c)*d^2*e^4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^2 + 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e^
4 + (a + 2*c)*d*e^5)*x)*sqrt(d^2 - e^2)*log((d^2*x + d*e + sqrt(d^2 - e^2)*(d*x + e) + (d^2 - e^2 + sqrt(d^2 -
e^2)*d)*sqrt(x^2 - 1))/(e*x + d)) + 2*(c*d^6*e + b*d^5*e^2 - (3*a + 5*c)*d^4*e^3 + b*d^3*e^4 + (3*a + 4*c)*d^
2*e^5 - 2*b*d*e^6)*x + (2*b*d^5*e^2 - (4*a + 3*c)*d^4*e^3 - b*d^3*e^4 + (5*a + 3*c)*d^2*e^5 - b*d*e^6 - a*e^7
+ (c*d^5*e^2 + b*d^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x)*sqrt(x^2 - 1))/(d
^8*e^2 - 3*d^6*e^4 + 3*d^4*e^6 - d^2*e^8 + (d^6*e^4 - 3*d^4*e^6 + 3*d^2*e^8 - e^10)*x^2 + 2*(d^7*e^3 - 3*d^5*e
^5 + 3*d^3*e^7 - d*e^9)*x), 1/2*(c*d^7 + b*d^6*e - (3*a + 5*c)*d^5*e^2 + b*d^4*e^3 + (3*a + 4*c)*d^3*e^4 - 2*b
*d^2*e^5 + (c*d^5*e^2 + b*d^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x^2 - 2*((2
*a + c)*d^4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4 + ((2*a + c)*d^2*e^4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^2 + 2*((
2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5)*x)*sqrt(-d^2 + e^2)*arctan(-(sqrt(-d^2 + e^2)*sqrt(x^2 - 1)*
e - sqrt(-d^2 + e^2)*(e*x + d))/(d^2 - e^2)) + 2*(c*d^6*e + b*d^5*e^2 - (3*a + 5*c)*d^4*e^3 + b*d^3*e^4 + (3*a
+ 4*c)*d^2*e^5 - 2*b*d*e^6)*x + (2*b*d^5*e^2 - (4*a + 3*c)*d^4*e^3 - b*d^3*e^4 + (5*a + 3*c)*d^2*e^5 - b*d*e^
6 - a*e^7 + (c*d^5*e^2 + b*d^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x)*sqrt(x^
2 - 1))/(d^8*e^2 - 3*d^6*e^4 + 3*d^4*e^6 - d^2*e^8 + (d^6*e^4 - 3*d^4*e^6 + 3*d^2*e^8 - e^10)*x^2 + 2*(d^7*e^3
- 3*d^5*e^5 + 3*d^3*e^7 - d*e^9)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b x + c x^{2}}{\sqrt{\left (x - 1\right ) \left (x + 1\right )} \left (d + e x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(e*x+d)**3/(x**2-1)**(1/2),x)

[Out]

Integral((a + b*x + c*x**2)/(sqrt((x - 1)*(x + 1))*(d + e*x)**3), x)

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Giac [B]  time = 1.16424, size = 724, normalized size = 3.71 \begin{align*} \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 (x - \sqrt{x^{2} - 1}\right )} e + d}{\sqrt{-d^{2} + e^{2}}}\right )}{{\left (d^{4} - 2 \, d^{2} e^{2} + e^{4}\right )} \sqrt{-d^{2} + e^{2}}} + \frac{2 \, c d^{4}{\left (x - \sqrt{x^{2} - 1}\right )}^{3} e + 2 \, c d^{5}{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 2 \, b d^{4}{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e - 2 \, a d^{2}{\left (x - \sqrt{x^{2} - 1}\right )}^{3} e^{3} - 5 \, c d^{2}{\left (x - \sqrt{x^{2} - 1}\right )}^{3} e^{3} - 6 \, a d^{3}{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{2} - 7 \, c d^{3}{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{2} + 2 \, c d^{4}{\left (x - \sqrt{x^{2} - 1}\right )} e + 3 \, b d{\left (x - \sqrt{x^{2} - 1}\right )}^{3} e^{4} + 5 \, b d^{2}{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{3} + 4 \, b d^{3}{\left (x - \sqrt{x^{2} - 1}\right )} e^{2} - a{\left (x - \sqrt{x^{2} - 1}\right )}^{3} e^{5} - 3 \, a d{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{4} - 4 \, c d{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{4} - 10 \, a d^{2}{\left (x - \sqrt{x^{2} - 1}\right )} e^{3} - 11 \, c d^{2}{\left (x - \sqrt{x^{2} - 1}\right )} e^{3} + c d^{3} e^{2} + 2 \, b{\left (x - \sqrt{x^{2} - 1}\right )}^{2} e^{5} + 5 \, b d{\left (x - \sqrt{x^{2} - 1}\right )} e^{4} + b d^{2} e^{3} + a{\left (x - \sqrt{x^{2} - 1}\right )} e^{5} - 3 \, a d e^{4} - 4 \, c d e^{4} + 2 \, b e^{5}}{{\left (d^{4} e^{2} - 2 \, d^{2} e^{4} + e^{6}\right )}{\left ({\left (x - \sqrt{x^{2} - 1}\right )}^{2} e + 2 \, d{\left (x - \sqrt{x^{2} - 1}\right )} + e\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*d^2 + c*d^2 - 3*b*d*e + a*e^2 + 2*c*e^2)*arctan(-((x - sqrt(x^2 - 1))*e + d)/sqrt(-d^2 + e^2))/((d^4 - 2*
d^2*e^2 + e^4)*sqrt(-d^2 + e^2)) + (2*c*d^4*(x - sqrt(x^2 - 1))^3*e + 2*c*d^5*(x - sqrt(x^2 - 1))^2 + 2*b*d^4*
(x - sqrt(x^2 - 1))^2*e - 2*a*d^2*(x - sqrt(x^2 - 1))^3*e^3 - 5*c*d^2*(x - sqrt(x^2 - 1))^3*e^3 - 6*a*d^3*(x -
sqrt(x^2 - 1))^2*e^2 - 7*c*d^3*(x - sqrt(x^2 - 1))^2*e^2 + 2*c*d^4*(x - sqrt(x^2 - 1))*e + 3*b*d*(x - sqrt(x^
2 - 1))^3*e^4 + 5*b*d^2*(x - sqrt(x^2 - 1))^2*e^3 + 4*b*d^3*(x - sqrt(x^2 - 1))*e^2 - a*(x - sqrt(x^2 - 1))^3*
e^5 - 3*a*d*(x - sqrt(x^2 - 1))^2*e^4 - 4*c*d*(x - sqrt(x^2 - 1))^2*e^4 - 10*a*d^2*(x - sqrt(x^2 - 1))*e^3 - 1
1*c*d^2*(x - sqrt(x^2 - 1))*e^3 + c*d^3*e^2 + 2*b*(x - sqrt(x^2 - 1))^2*e^5 + 5*b*d*(x - sqrt(x^2 - 1))*e^4 +
b*d^2*e^3 + a*(x - sqrt(x^2 - 1))*e^5 - 3*a*d*e^4 - 4*c*d*e^4 + 2*b*e^5)/((d^4*e^2 - 2*d^2*e^4 + e^6)*((x - sq
rt(x^2 - 1))^2*e + 2*d*(x - sqrt(x^2 - 1)) + e)^2)