### 3.2352 $$\int \frac{(a+b x+c x^2)^{3/2}}{(d+e x)^6} \, dx$$

Optimal. Leaf size=296 $-\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{16 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}$

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(128*(c*d^2 - b*d*e + a
*e^2)^3*(d + e*x)^2) + ((2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(16*(c*d^2 - b*
d*e + a*e^2)^2*(d + e*x)^4) - (e*(a + b*x + c*x^2)^(5/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) + (3*(b^2 -
4*a*c)^2*(2*c*d - b*e)*ArcTanh[(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])])/(256*(c*d^2 - b*d*e + a*e^2)^(7/2))

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Rubi [A]  time = 0.234795, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {730, 720, 724, 206} $-\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{16 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(128*(c*d^2 - b*d*e + a
*e^2)^3*(d + e*x)^2) + ((2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(16*(c*d^2 - b*
d*e + a*e^2)^2*(d + e*x)^4) - (e*(a + b*x + c*x^2)^(5/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) + (3*(b^2 -
4*a*c)^2*(2*c*d - b*e)*ArcTanh[(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])])/(256*(c*d^2 - b*d*e + a*e^2)^(7/2))

Rule 730

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(2*c*d - b*e)/(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, m, p}, 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 + 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{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx &=-\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac{(2 c d-b e) \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac{\left (3 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx}{32 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac{(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac{\left (3 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{256 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac{(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}-\frac{\left (3 \left (b^2-4 a c\right )^2 (2 c d-b 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 )}{128 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{3 \left (b^2-4 a c\right ) (2 c d-b e) (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2}}{128 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac{(2 c d-b e) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{16 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^5}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b 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 )}{256 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.25351, size = 275, normalized size = 0.93 $-\frac{(2 c d-b e) \left (3 \left (b^2-4 a c\right ) \left (\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{\sqrt{a+x (b+c x)} (-2 a e+b (d-e x)+2 c d x)}{4 (d+e x)^2 \left (e (a e-b d)+c d^2\right )}\right )+\frac{2 (a+x (b+c x))^{3/2} (2 a e-b d+b e x-2 c d x)}{(d+e x)^4}\right )}{32 \left (e (a e-b d)+c d^2\right )^2}-\frac{e (a+x (b+c x))^{5/2}}{5 (d+e x)^5 \left (e (a e-b d)+c d^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

-(e*(a + x*(b + c*x))^(5/2))/(5*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^5) - ((2*c*d - b*e)*((2*(-(b*d) + 2*a*e -
2*c*d*x + b*e*x)*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 + 3*(b^2 - 4*a*c)*((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)))
))/(32*(c*d^2 + e*(-(b*d) + a*e))^2)

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Maple [B]  time = 0.245, size = 20477, normalized size = 69.2 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(e*x+d)^6,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((c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**6,x)

[Out]

Timed out

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Giac [B]  time = 4.63398, size = 11048, normalized size = 37.32 \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)^(3/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

3/128*(2*b^4*c*d - 16*a*b^2*c^2*d + 32*a^2*c^3*d - b^5*e + 8*a*b^3*c*e - 16*a^2*b*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^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e
^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3
*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/640*(2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*c^(13/2)*d^8*e + 1024*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^7*d^9 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*c^6*d^7*e^2 + 3072*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^6*d^8*e + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(13/2)*d^9
+ 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*c^(11/2)*d^6*e^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b*c
^(11/2)*d^7*e^2 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(11/2)*d^8*e - 2560*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^4*a*c^(13/2)*d^8*e + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^6*d^9 - 3840*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^7*b*c^5*d^6*e^3 - 7936*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^2*c^5*d^7*e^2 - 512*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^6*d^7*e^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c^5*d^8*e - 51
20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^6*d^8*e + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(11/
2)*d^9 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b*c^(9/2)*d^5*e^4 - 6400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^6*b^2*c^(9/2)*d^6*e^3 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*c^(11/2)*d^6*e^3 - 6400*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^4*b^3*c^(9/2)*d^7*e^2 + 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(11/2)*d^7*e^2
- 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^4*c^(9/2)*d^8*e - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*
b^2*c^(11/2)*d^8*e + 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c^5*d^9 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^7*b^2*c^4*d^5*e^4 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*c^5*d^5*e^4 - 1280*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^5*b^3*c^4*d^6*e^3 + 24832*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b*c^5*d^6*e^3 - 1600*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^3*b^4*c^4*d^7*e^2 + 17920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^2*c^5*d^7*e^2
+ 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^6*d^7*e^2 - 640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*c^
4*d^8*e - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c^5*d^8*e + 32*b^5*c^(9/2)*d^9 + 3840*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^8*b^2*c^(7/2)*d^4*e^5 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*c^(9/2)*d^4*e^5 + 12
80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^3*c^(7/2)*d^5*e^4 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b*
c^(9/2)*d^5*e^4 + 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*c^(7/2)*d^6*e^3 + 19200*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^4*a*b^2*c^(9/2)*d^6*e^3 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(11/2)*d^6*e^3 + 160
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^5*c^(7/2)*d^7*e^2 + 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^3
*c^(9/2)*d^7*e^2 + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^(11/2)*d^7*e^2 - 64*b^6*c^(7/2)*d^8*e -
160*a*b^4*c^(9/2)*d^8*e + 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^3*c^3*d^4*e^5 - 3840*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^7*a*b*c^4*d^4*e^5 + 4280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^4*c^3*d^5*e^4 - 18880*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c^4*d^5*e^4 - 25216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^5*d^5*e
^4 + 2080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c^3*d^6*e^3 - 24320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*
a^2*b*c^5*d^6*e^3 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^6*c^3*d^7*e^2 + 4000*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))*a*b^4*c^4*d^7*e^2 + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c^5*d^7*e^2 - 1280*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^8*b^3*c^(5/2)*d^3*e^6 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b*c^(7/2)*d^3*e^6
+ 7420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^4*c^(5/2)*d^4*e^5 - 3040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*
a*b^2*c^(7/2)*d^4*e^5 - 16960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*c^(9/2)*d^4*e^5 + 2860*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^4*b^5*c^(5/2)*d^5*e^4 - 21600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^3*c^(7/2)*d^5*e^4
- 40000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b*c^(9/2)*d^5*e^4 + 860*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^2*b^6*c^(5/2)*d^6*e^3 - 5200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^(7/2)*d^6*e^3 - 24000*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^(9/2)*d^6*e^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*c^(11/2)*d^
6*e^3 + 12*b^7*c^(5/2)*d^7*e^2 + 464*a*b^5*c^(7/2)*d^7*e^2 + 320*a^2*b^3*c^(9/2)*d^7*e^2 - 4780*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^7*b^4*c^2*d^3*e^6 - 7840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^2*c^3*d^3*e^6 - 7360*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*c^4*d^3*e^6 + 1448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^5*c^2*d^4
*e^5 + 8640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^3*c^3*d^4*e^5 + 12160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^5*a^2*b*c^4*d^4*e^5 + 540*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^6*c^2*d^5*e^4 - 7520*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))^3*a*b^4*c^3*d^5*e^4 - 13120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*c^4*d^5*e^4 + 12800*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*c^5*d^5*e^4 + 200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^7*c^2*d^6*e^3
- 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c^3*d^6*e^3 - 9600*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b
^3*c^4*d^6*e^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c^5*d^6*e^3 - 270*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^8*b^4*c^(3/2)*d^2*e^7 + 6000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^2*c^(5/2)*d^2*e^7 - 480*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^8*a^2*c^(7/2)*d^2*e^7 - 5330*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^5*c^(3/2)*d^
3*e^6 - 9840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^3*c^(5/2)*d^3*e^6 + 8160*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^6*a^2*b*c^(7/2)*d^3*e^6 - 1390*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^6*c^(3/2)*d^4*e^5 + 9620*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^4*a*b^4*c^(5/2)*d^4*e^5 + 37120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^2*c^(7
/2)*d^4*e^5 + 37440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*c^(9/2)*d^4*e^5 - 230*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^2*b^7*c^(3/2)*d^5*e^4 - 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^5*c^(5/2)*d^5*e^4 + 5920*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^3*c^(7/2)*d^5*e^4 + 23040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b*c
^(9/2)*d^5*e^4 + 20*b^8*c^(3/2)*d^6*e^3 - 204*a*b^6*c^(5/2)*d^6*e^3 - 1360*a^2*b^4*c^(7/2)*d^6*e^3 - 320*a^3*b
^2*c^(9/2)*d^6*e^3 - 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^4*c*d*e^8 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^9*a*b^2*c^2*d*e^8 - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*c^3*d*e^8 + 330*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^7*b^5*c*d^2*e^7 + 8880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^3*c^2*d^2*e^7 + 9120*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^7*a^2*b*c^3*d^2*e^7 - 2626*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^6*c*d^3*e^6 - 626
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^4*c^2*d^3*e^6 - 21760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^
2*c^3*d^3*e^6 + 29120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*c^4*d^3*e^6 - 930*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^3*b^7*c*d^4*e^5 + 4760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^5*c^2*d^4*e^5 + 14240*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^3*a^2*b^3*c^3*d^4*e^5 + 42880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b*c^4*d^4*e^5 - 1
20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^8*c*d^5*e^4 + 260*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^6*c^2*d^5*e
^4 + 3880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^4*c^3*d^5*e^4 + 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*
a^3*b^2*c^4*d^5*e^4 + 640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*c^5*d^5*e^4 + 135*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^8*b^5*sqrt(c)*d*e^8 - 1080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^3*c^(3/2)*d*e^8 - 1680*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^8*a^2*b*c^(5/2)*d*e^8 + 490*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^6*sqrt(c)*d^2*
e^7 + 12420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^4*c^(3/2)*d^2*e^7 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^6*a^2*b^2*c^(5/2)*d^2*e^7 + 18240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*c^(7/2)*d^2*e^7 - 640*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^4*b^7*sqrt(c)*d^3*e^6 + 1390*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^5*c^(3/2)*
d^3*e^6 - 41840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^3*c^(5/2)*d^3*e^6 - 2080*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^4*a^3*b*c^(7/2)*d^3*e^6 - 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^8*sqrt(c)*d^4*e^5 + 1720*(sqr
t(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^6*c^(3/2)*d^4*e^5 - 2380*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^4*c
^(5/2)*d^4*e^5 + 10720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^2*c^(7/2)*d^4*e^5 - 7360*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^2*a^4*c^(9/2)*d^4*e^5 - 15*b^9*sqrt(c)*d^5*e^4 + 30*a*b^7*c^(3/2)*d^5*e^4 + 532*a^2*b^5*c^(5
/2)*d^5*e^4 + 2240*a^3*b^3*c^(7/2)*d^5*e^4 + 320*a^4*b*c^(9/2)*d^5*e^4 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^9*b^5*e^9 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c*e^9 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
9*a^2*b*c^2*e^9 + 70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^6*d*e^8 - 420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^7*a*b^4*c*d*e^8 - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^2*c^2*d*e^8 + 4800*(sqrt(c)*x - sqrt(c*x^
2 + b*x + a))^7*a^3*c^3*d*e^8 + 128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^7*d^2*e^7 + 9026*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^5*a*b^5*c*d^2*e^7 + 1520*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^3*c^2*d^2*e^7 + 11040*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b*c^3*d^2*e^7 - 70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^8*d^3*e^6
+ 2280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^6*c*d^3*e^6 - 20420*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^
2*b^4*c^2*d^3*e^6 - 20320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^2*c^3*d^3*e^6 - 24640*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^3*a^4*c^4*d^3*e^6 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^9*d^4*e^5 + 450*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))*a*b^7*c*d^4*e^5 - 2080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^5*c^2*d^4*e^5 - 1120*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^3*c^3*d^4*e^5 - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b*c^4*d^4*
e^5 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^3*c^(5/2)*e^9 - 490*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a
*b^5*sqrt(c)*d*e^8 - 11440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^3*c^(3/2)*d*e^8 - 1440*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))^6*a^3*b*c^(5/2)*d*e^8 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^6*sqrt(c)*d^2*e^7 +
5120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^4*c^(3/2)*d^2*e^7 + 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^4*a^3*b^2*c^(5/2)*d^2*e^7 - 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*c^(7/2)*d^2*e^7 + 630*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*a*b^7*sqrt(c)*d^3*e^6 - 4430*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^5*c^(3/2)
*d^3*e^6 - 2480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^3*c^(5/2)*d^3*e^6 - 22240*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*a^4*b*c^(7/2)*d^3*e^6 + 60*a*b^8*sqrt(c)*d^4*e^5 - 330*a^2*b^6*c^(3/2)*d^4*e^5 - 260*a^3*b^4*c^(
5/2)*d^4*e^5 - 2560*a^4*b^2*c^(7/2)*d^4*e^5 - 64*a^5*c^(9/2)*d^4*e^5 - 70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
7*a*b^5*e^9 + 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b^3*c*e^9 + 2720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^7*a^3*b*c^2*e^9 - 256*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^6*d*e^8 - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^5*a^2*b^4*c*d*e^8 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^4*c^3*d*e^8 + 210*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^3*a*b^7*d^2*e^7 + 230*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^5*c*d^2*e^7 + 26480*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^3*c^2*d^2*e^7 + 6240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b*c^3*d^2*
e^7 + 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^8*d^3*e^6 - 750*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^6*c
*d^3*e^6 + 2820*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^4*c^2*d^3*e^6 - 5760*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a^4*b^2*c^3*d^3*e^6 + 2880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*c^4*d^3*e^6 + 5120*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^6*a^3*b^2*c^(3/2)*e^9 - 3200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^5*sqrt(c)*d*e^8 - 76
80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^3*c^(3/2)*d*e^8 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^
4*b*c^(5/2)*d*e^8 - 630*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^6*sqrt(c)*d^2*e^7 + 7180*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^2*a^3*b^4*c^(3/2)*d^2*e^7 + 7360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^2*c^(5/2)*d^2*
e^7 + 11200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*c^(7/2)*d^2*e^7 - 90*a^2*b^7*sqrt(c)*d^3*e^6 + 610*a^3*b
^5*c^(3/2)*d^3*e^6 - 560*a^4*b^3*c^(5/2)*d^3*e^6 + 1568*a^5*b*c^(7/2)*d^3*e^6 + 128*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))^5*a^2*b^5*e^9 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^3*c*e^9 + 3840*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^5*a^4*b*c^2*e^9 - 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^6*d*e^8 - 3580*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^3*a^3*b^4*c*d*e^8 - 13760*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b^2*c^2*d*e^8 + 544
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*c^3*d*e^8 - 90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^7*d^2*e^7
+ 750*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^5*c*d^2*e^7 - 400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^3
*c^2*d^2*e^7 + 6880*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^5*b*c^3*d^2*e^7 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*a^3*b^4*sqrt(c)*e^9 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*b^2*c^(3/2)*e^9 + 2560*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^4*a^5*c^(5/2)*e^9 + 210*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^5*sqrt(c)*d*e^8
- 6800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^3*c^(3/2)*d*e^8 - 3040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^2*a^5*b*c^(5/2)*d*e^8 + 60*a^3*b^6*sqrt(c)*d^2*e^7 - 450*a^4*b^4*c^(3/2)*d^2*e^7 + 976*a^5*b^2*c^(5/2)*d^2*e^
7 - 288*a^6*c^(7/2)*d^2*e^7 + 70*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3*b^5*e^9 + 2000*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^3*a^4*b^3*c*e^9 + 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^5*b*c^2*e^9 + 60*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*a^3*b^6*d*e^8 - 450*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^4*c*d*e^8 - 1840*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))*a^5*b^2*c^2*d*e^8 - 2080*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*c^3*d*e^8 + 2560*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^2*c^(3/2)*e^9 - 15*a^4*b^5*sqrt(c)*d*e^8 + 120*a^5*b^3*c^(3/2)*d*e^
8 - 752*a^6*b*c^(5/2)*d*e^8 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^5*e^9 + 120*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*a^5*b^3*c*e^9 + 1040*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b*c^2*e^9 + 256*a^7*c^(5/2)*e^9)/((c
^3*d^6*e^4 - 3*b*c^2*d^5*e^5 + 3*b^2*c*d^4*e^6 + 3*a*c^2*d^4*e^6 - b^3*d^3*e^7 - 6*a*b*c*d^3*e^7 + 3*a*b^2*d^2
*e^8 + 3*a^2*c*d^2*e^8 - 3*a^2*b*d*e^9 + a^3*e^10)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^5)