3.1554 \(\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=217 \[ -\frac{e \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{e \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )} \]

[Out]

-((b^2 - 4*a*c)*e*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x
)^2) + ((2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((b^2 - 4*a*c)^2*e*Ar
cTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(16*(c*d^2 - b*d
*e + a*e^2)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.226268, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {806, 720, 724, 206} \[ -\frac{e \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{e \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2 - 4*a*c)*e*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x
)^2) + ((2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((b^2 - 4*a*c)^2*e*Ar
cTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(16*(c*d^2 - b*d
*e + a*e^2)^(5/2))

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

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{(b+2 c x) \sqrt{a+b x+c x^2}}{(d+e x)^4} \, dx &=\frac{(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\left (\left (b^2-4 a c\right ) e\right ) \int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{\left (b^2-4 a c\right ) e (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (\left (b^2-4 a c\right )^2 e\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{16 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{\left (b^2-4 a c\right ) e (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\left (\left (b^2-4 a c\right )^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{\left (b^2-4 a c\right ) e (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{8 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (b^2-4 a c\right )^2 e \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.532198, size = 214, normalized size = 0.99 \[ \frac{\frac{2 (a+x (b+c x))^{3/2} (2 c d-b e)}{(d+e x)^3}-3 e \left (b^2-4 a c\right ) \left (\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}\right )}{6 \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 - 3*(b^2 - 4*a*c)*e*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2
*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2
*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)
)))/(6*(c*d^2 + e*(-(b*d) + a*e)))

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Maple [B]  time = 0.019, size = 7916, normalized size = 36.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 38.4673, size = 4239, normalized size = 19.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(3*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4*x^3 + 3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d*e^3*x^2 + 3*(b^4 - 8*a*b
^2*c + 16*a^2*c^2)*d^2*e^2*x + (b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^3*e)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*
e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a
*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*
x)/(e^2*x^2 + 2*d*e*x + d^2)) + 4*(16*a*c^3*d^5 - 8*a^3*b*e^5 - (3*b^3*c + 28*a*b*c^2)*d^4*e + (3*b^4 + 26*a*b
^2*c + 8*a^2*c^2)*d^3*e^2 - (17*a*b^3 + 12*a^2*b*c)*d^2*e^3 + 2*(11*a^2*b^2 - 4*a^3*c)*d*e^4 + (16*c^4*d^5 - 4
0*b*c^3*d^4*e + 2*(13*b^2*c^2 + 28*a*c^3)*d^3*e^2 + (b^3*c - 84*a*b*c^2)*d^2*e^3 - (3*b^4 - 22*a*b^2*c - 40*a^
2*c^2)*d*e^4 + (3*a*b^3 - 20*a^2*b*c)*e^5)*x^2 + 2*(8*b*c^3*d^5 - (23*b^2*c^2 - 12*a*c^3)*d^4*e + (19*b^3*c +
4*a*b*c^2)*d^3*e^2 - 4*(b^4 + 6*a*b^2*c)*d^2*e^3 + 5*(a*b^3 + 4*a^2*b*c)*d*e^4 - (a^2*b^2 + 12*a^3*c)*e^5)*x)*
sqrt(c*x^2 + b*x + a))/(c^3*d^9 - 3*b*c^2*d^8*e - 3*a^2*b*d^4*e^5 + a^3*d^3*e^6 + 3*(b^2*c + a*c^2)*d^7*e^2 -
(b^3 + 6*a*b*c)*d^6*e^3 + 3*(a*b^2 + a^2*c)*d^5*e^4 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 - 3*a^2*b*d*e^8 + a^3*e^9
+ 3*(b^2*c + a*c^2)*d^4*e^5 - (b^3 + 6*a*b*c)*d^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^7)*x^3 + 3*(c^3*d^7*e^2 - 3*b
*c^2*d^6*e^3 - 3*a^2*b*d^2*e^7 + a^3*d*e^8 + 3*(b^2*c + a*c^2)*d^5*e^4 - (b^3 + 6*a*b*c)*d^4*e^5 + 3*(a*b^2 +
a^2*c)*d^3*e^6)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 - 3*a^2*b*d^3*e^6 + a^3*d^2*e^7 + 3*(b^2*c + a*c^2)*d^6*e
^3 - (b^3 + 6*a*b*c)*d^5*e^4 + 3*(a*b^2 + a^2*c)*d^4*e^5)*x), 1/48*(3*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4*x^3
+ 3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d*e^3*x^2 + 3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^2*e^2*x + (b^4 - 8*a*b^2*c +
16*a^2*c^2)*d^3*e)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a
)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2
- b^2*d*e + a*b*e^2)*x)) + 2*(16*a*c^3*d^5 - 8*a^3*b*e^5 - (3*b^3*c + 28*a*b*c^2)*d^4*e + (3*b^4 + 26*a*b^2*c
+ 8*a^2*c^2)*d^3*e^2 - (17*a*b^3 + 12*a^2*b*c)*d^2*e^3 + 2*(11*a^2*b^2 - 4*a^3*c)*d*e^4 + (16*c^4*d^5 - 40*b*c
^3*d^4*e + 2*(13*b^2*c^2 + 28*a*c^3)*d^3*e^2 + (b^3*c - 84*a*b*c^2)*d^2*e^3 - (3*b^4 - 22*a*b^2*c - 40*a^2*c^2
)*d*e^4 + (3*a*b^3 - 20*a^2*b*c)*e^5)*x^2 + 2*(8*b*c^3*d^5 - (23*b^2*c^2 - 12*a*c^3)*d^4*e + (19*b^3*c + 4*a*b
*c^2)*d^3*e^2 - 4*(b^4 + 6*a*b^2*c)*d^2*e^3 + 5*(a*b^3 + 4*a^2*b*c)*d*e^4 - (a^2*b^2 + 12*a^3*c)*e^5)*x)*sqrt(
c*x^2 + b*x + a))/(c^3*d^9 - 3*b*c^2*d^8*e - 3*a^2*b*d^4*e^5 + a^3*d^3*e^6 + 3*(b^2*c + a*c^2)*d^7*e^2 - (b^3
+ 6*a*b*c)*d^6*e^3 + 3*(a*b^2 + a^2*c)*d^5*e^4 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 - 3*a^2*b*d*e^8 + a^3*e^9 + 3*
(b^2*c + a*c^2)*d^4*e^5 - (b^3 + 6*a*b*c)*d^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^7)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*
d^6*e^3 - 3*a^2*b*d^2*e^7 + a^3*d*e^8 + 3*(b^2*c + a*c^2)*d^5*e^4 - (b^3 + 6*a*b*c)*d^4*e^5 + 3*(a*b^2 + a^2*c
)*d^3*e^6)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 - 3*a^2*b*d^3*e^6 + a^3*d^2*e^7 + 3*(b^2*c + a*c^2)*d^6*e^3 -
(b^3 + 6*a*b*c)*d^5*e^4 + 3*(a*b^2 + a^2*c)*d^4*e^5)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}}}{\left (d + e x\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.71163, size = 3525, normalized size = 16.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b^4*e - 8*a*b^2*c*e + 16*a^2*c^2*e)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d
^2 + b*d*e - a*e^2))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(-c*d^
2 + b*d*e - a*e^2)) + 1/24*(192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*c^(9/2)*d^5*e + 128*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^3*c^5*d^6 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^4*d^4*e^2 + 192*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*b*c^(9/2)*d^6 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(7/2)*d^4*e^2 - 240*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*b^2*c^(7/2)*d^5*e - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(9/2)*d^5*e + 96*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c^4*d^6 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^3*d^3*e^3 - 272*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^3*d^4*e^2 + 128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^4*d^4*e^
2 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c^3*d^5*e - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^4*d^
5*e + 16*b^3*c^(7/2)*d^6 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(5/2)*d^3*e^3 + 384*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^4*a*c^(7/2)*d^3*e^3 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(5/2)*d^4*e^2 + 672*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^(7/2)*d^4*e^2 - 24*b^4*c^(5/2)*d^5*e - 48*a*b^2*c^(7/2)*d^5*e + 96*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^2*d^2*e^4 + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^3*d^2*e^
4 + 32*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c^2*d^3*e^3 + 512*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c
^3*d^3*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c^2*d^4*e^2 + 528*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*
a*b^2*c^3*d^4*e^2 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c^4*d^4*e^2 + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*b^3*c^(3/2)*d^2*e^4 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(5/2)*d^2*e^4 + 42*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*b^4*c^(3/2)*d^3*e^3 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(5/2)*d^3*e^3
- 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(7/2)*d^3*e^3 + 2*b^5*c^(3/2)*d^4*e^2 + 112*a*b^3*c^(5/2)*d^
4*e^2 + 48*a^2*b*c^(7/2)*d^4*e^2 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c^2*d*e^5 + 90*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^3*b^4*c*d^2*e^4 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c^2*d^2*e^4 - 480*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^3*d^2*e^4 + 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*c*d^3*e^3 - 144
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c^2*d^3*e^3 - 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^3*d^3
*e^3 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*sqrt(c)*d*e^5 - 168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*
a*b^2*c^(3/2)*d*e^5 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(5/2)*d*e^5 + 24*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*b^5*sqrt(c)*d^2*e^4 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3*c^(3/2)*d^2*e^4 + 3*b^6*sqr
t(c)*d^3*e^3 - 26*a*b^4*c^(3/2)*d^3*e^3 - 192*a^2*b^2*c^(5/2)*d^3*e^3 - 32*a^3*c^(7/2)*d^3*e^3 - 3*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*b^4*e^6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c*e^6 + 48*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^5*a^2*c^2*e^6 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*d*e^5 - 160*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^3*a*b^3*c*d*e^5 + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c^2*d*e^5 + 3*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*b^6*d^2*e^4 - 42*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*c*d^2*e^4 + 288*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*a^2*b^2*c^2*d^2*e^4 + 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c^3*d^2*e^4 + 144*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b*c^(3/2)*e^6 - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*sqrt(
c)*d*e^5 + 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^(3/2)*d*e^5 + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*a^3*c^(5/2)*d*e^5 - 6*a*b^5*sqrt(c)*d^2*e^4 + 48*a^2*b^3*c^(3/2)*d^2*e^4 + 192*a^3*b*c^(5/2)*d^2*e^4
+ 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^4*e^6 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*c*e^6 -
6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*d*e^5 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c^2*d*e^5 +
48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^3*sqrt(c)*e^6 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b*
c^(3/2)*e^6 + 3*a^2*b^4*sqrt(c)*d*e^5 - 40*a^3*b^2*c^(3/2)*d*e^5 - 80*a^4*c^(5/2)*d*e^5 + 3*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))*a^2*b^4*e^6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^2*c*e^6 - 48*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))*a^4*c^2*e^6 + 16*a^4*b*c^(3/2)*e^6)/((c^2*d^4*e^3 - 2*b*c*d^3*e^4 + b^2*d^2*e^5 + 2*a*c*d^2*e^
5 - 2*a*b*d*e^6 + a^2*e^7)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sq
rt(c)*d + b*d - a*e)^3)