### 3.1250 $$\int \frac{(b d+2 c d x)^6}{(a+b x+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=136 $40 c^{3/2} d^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )+80 c^2 d^6 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}$

[Out]

(-2*d^6*(b + 2*c*x)^5)/(3*(a + b*x + c*x^2)^(3/2)) - (40*c*d^6*(b + 2*c*x)^3)/(3*Sqrt[a + b*x + c*x^2]) + 80*c
^2*d^6*(b + 2*c*x)*Sqrt[a + b*x + c*x^2] + 40*c^(3/2)*(b^2 - 4*a*c)*d^6*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])]

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Rubi [A]  time = 0.0771676, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {686, 692, 621, 206} $40 c^{3/2} d^6 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )+80 c^2 d^6 (b+2 c x) \sqrt{a+b x+c x^2}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*d^6*(b + 2*c*x)^5)/(3*(a + b*x + c*x^2)^(3/2)) - (40*c*d^6*(b + 2*c*x)^3)/(3*Sqrt[a + b*x + c*x^2]) + 80*c
^2*d^6*(b + 2*c*x)*Sqrt[a + b*x + c*x^2] + 40*c^(3/2)*(b^2 - 4*a*c)*d^6*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])]

Rule 686

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*(d + e*x)^(m - 1)*
(a + b*x + c*x^2)^(p + 1))/(b*(p + 1)), x] - Dist[(d*e*(m - 1))/(b*(p + 1)), 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] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2
*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 d+2 c d x)^6}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac{1}{3} \left (20 c d^2\right ) \int \frac{(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}+\left (80 c^2 d^4\right ) \int \frac{(b d+2 c d x)^2}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}+80 c^2 d^6 (b+2 c x) \sqrt{a+b x+c x^2}+\left (40 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}+80 c^2 d^6 (b+2 c x) \sqrt{a+b x+c x^2}+\left (80 c^2 \left (b^2-4 a c\right ) d^6\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )\\ &=-\frac{2 d^6 (b+2 c x)^5}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{40 c d^6 (b+2 c x)^3}{3 \sqrt{a+b x+c x^2}}+80 c^2 d^6 (b+2 c x) \sqrt{a+b x+c x^2}+40 c^{3/2} \left (b^2-4 a c\right ) d^6 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 1.50591, size = 204, normalized size = 1.5 $\frac{d^6 \left (3 (b+2 c x)^5-\frac{5 \left (b^2-4 a c\right ) \left (\sqrt{4 a-\frac{b^2}{c}} (b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (4 c \left (3 a+4 c x^2\right )+b^2+16 b c x\right )-24 c^{3/2} (a+x (b+c x))^2 \sinh ^{-1}\left (\frac{b+2 c x}{\sqrt{c} \sqrt{4 a-\frac{b^2}{c}}}\right )\right )}{\sqrt{4 a-\frac{b^2}{c}} \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}}\right )}{3 (a+x (b+c x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^(5/2),x]

[Out]

(d^6*(3*(b + 2*c*x)^5 - (5*(b^2 - 4*a*c)*(Sqrt[4*a - b^2/c]*(b + 2*c*x)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a
*c)]*(b^2 + 16*b*c*x + 4*c*(3*a + 4*c*x^2)) - 24*c^(3/2)*(a + x*(b + c*x))^2*ArcSinh[(b + 2*c*x)/(Sqrt[4*a - b
^2/c]*Sqrt[c])]))/(Sqrt[4*a - b^2/c]*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/(3*(a + x*(b + c*x))^(3/2))

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Maple [B]  time = 0.063, size = 997, normalized size = 7.3 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^(5/2),x)

[Out]

40*d^6*c^(3/2)*b^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+15/2*d^6*b^7/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+20
*d^6*c*b^3/(c*x^2+b*x+a)^(1/2)+32*d^6*c^5*x^5/(c*x^2+b*x+a)^(3/2)-160*d^6*c^(5/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))+120*d^6*c^2*b^6/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-60*d^6*c*b^5*a/(4*a*c-b^2)/(c*x^2+b*x+a)
^(3/2)-480*d^6*c^2*b^5*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+160/3*d^6*c^4*a*x^3/(c*x^2+b*x+a)^(3/2)+160*d^6*c^3
*a*x/(c*x^2+b*x+a)^(1/2)-80*d^6*c^2*a*b/(c*x^2+b*x+a)^(1/2)+160*d^6*c^2*b*a^2/(c*x^2+b*x+a)^(3/2)-40/3*d^6*c^3
*b^2*x^3/(c*x^2+b*x+a)^(3/2)-140*d^6*c^2*b^3*x^2/(c*x^2+b*x+a)^(3/2)-65*d^6*c*b^4*x/(c*x^2+b*x+a)^(3/2)+60*d^6
*c*b^7/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-310/3*d^6*c*b^3*a/(c*x^2+b*x+a)^(3/2)-40*d^6*c^2*b^2*x/(c*x^2+b*x+a)^
(1/2)+20*d^6*c*b^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+80*d^6*c^4*b*x^4/(c*x^2+b*x+a)^(3/2)+240*d^6*c^3*b*a*x^2/(c
*x^2+b*x+a)^(3/2)+60*d^6*c^2*b^2*a*x/(c*x^2+b*x+a)^(3/2)+120*d^6*c^2*b^3*a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+9
60*d^6*c^3*b^3*a^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+15*d^6*c*b^6/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+40*d^6*c^2
*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-80*d^6*c^2*a*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+41/6*d^6*b^5/(c*x^2+b*
x+a)^(3/2)-960*d^6*c^3*b^4*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-160*d^6*c^3*a*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(
1/2)*x+240*d^6*c^3*b^2*a^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+1920*d^6*c^4*b^2*a^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^
(1/2)*x-120*d^6*c^2*b^4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x

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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*d*x+b*d)^6/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 11.5415, size = 1482, normalized size = 10.9 \begin{align*} \left [-\frac{2 \,{\left (30 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} +{\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \,{\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) -{\left (48 \, c^{5} d^{6} x^{5} + 120 \, b c^{4} d^{6} x^{4} + 40 \,{\left (b^{2} c^{3} + 8 \, a c^{4}\right )} d^{6} x^{3} - 60 \,{\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{6} x^{2} - 30 \,{\left (b^{4} c - 4 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x -{\left (b^{5} + 20 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}, -\frac{2 \,{\left (60 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} +{\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \,{\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x +{\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) -{\left (48 \, c^{5} d^{6} x^{5} + 120 \, b c^{4} d^{6} x^{4} + 40 \,{\left (b^{2} c^{3} + 8 \, a c^{4}\right )} d^{6} x^{3} - 60 \,{\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{6} x^{2} - 30 \,{\left (b^{4} c - 4 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x -{\left (b^{5} + 20 \, a b^{3} c - 120 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt{c x^{2} + b x + a}\right )}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-2/3*(30*((b^2*c^3 - 4*a*c^4)*d^6*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*d^6*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d
^6*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*d^6*x + (a^2*b^2*c - 4*a^3*c^2)*d^6)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2
+ 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - (48*c^5*d^6*x^5 + 120*b*c^4*d^6*x^4 + 40*(b^2*c^3 +
8*a*c^4)*d^6*x^3 - 60*(b^3*c^2 - 8*a*b*c^3)*d^6*x^2 - 30*(b^4*c - 4*a*b^2*c^2 - 8*a^2*c^3)*d^6*x - (b^5 + 20*a
*b^3*c - 120*a^2*b*c^2)*d^6)*sqrt(c*x^2 + b*x + a))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2),
-2/3*(60*((b^2*c^3 - 4*a*c^4)*d^6*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*d^6*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d
^6*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*d^6*x + (a^2*b^2*c - 4*a^3*c^2)*d^6)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x +
a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - (48*c^5*d^6*x^5 + 120*b*c^4*d^6*x^4 + 40*(b^2*c^3 + 8*a*c^
4)*d^6*x^3 - 60*(b^3*c^2 - 8*a*b*c^3)*d^6*x^2 - 30*(b^4*c - 4*a*b^2*c^2 - 8*a^2*c^3)*d^6*x - (b^5 + 20*a*b^3*c
- 120*a^2*b*c^2)*d^6)*sqrt(c*x^2 + b*x + a))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)]

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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((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.20005, size = 713, normalized size = 5.24 \begin{align*} -\frac{40 \,{\left (b^{2} c^{2} d^{6} - 4 \, a c^{3} d^{6}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{\sqrt{c}} + \frac{2 \,{\left (2 \,{\left (2 \,{\left (2 \,{\left (3 \,{\left (\frac{2 \,{\left (b^{4} c^{8} d^{6} - 8 \, a b^{2} c^{9} d^{6} + 16 \, a^{2} c^{10} d^{6}\right )} x}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}} + \frac{5 \,{\left (b^{5} c^{7} d^{6} - 8 \, a b^{3} c^{8} d^{6} + 16 \, a^{2} b c^{9} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x + \frac{5 \,{\left (b^{6} c^{6} d^{6} - 48 \, a^{2} b^{2} c^{8} d^{6} + 128 \, a^{3} c^{9} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac{15 \,{\left (b^{7} c^{5} d^{6} - 16 \, a b^{5} c^{6} d^{6} + 80 \, a^{2} b^{3} c^{7} d^{6} - 128 \, a^{3} b c^{8} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac{15 \,{\left (b^{8} c^{4} d^{6} - 12 \, a b^{6} c^{5} d^{6} + 40 \, a^{2} b^{4} c^{6} d^{6} - 128 \, a^{4} c^{8} d^{6}\right )}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )} x - \frac{b^{9} c^{3} d^{6} + 12 \, a b^{7} c^{4} d^{6} - 264 \, a^{2} b^{5} c^{5} d^{6} + 1280 \, a^{3} b^{3} c^{6} d^{6} - 1920 \, a^{4} b c^{7} d^{6}}{b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

-40*(b^2*c^2*d^6 - 4*a*c^3*d^6)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/sqrt(c) + 2/3*(2*
(2*(2*(3*(2*(b^4*c^8*d^6 - 8*a*b^2*c^9*d^6 + 16*a^2*c^10*d^6)*x/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5) + 5*(b^5*
c^7*d^6 - 8*a*b^3*c^8*d^6 + 16*a^2*b*c^9*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x + 5*(b^6*c^6*d^6 - 48*a^
2*b^2*c^8*d^6 + 128*a^3*c^9*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - 15*(b^7*c^5*d^6 - 16*a*b^5*c^6*d^6
+ 80*a^2*b^3*c^7*d^6 - 128*a^3*b*c^8*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - 15*(b^8*c^4*d^6 - 12*a*b^6
*c^5*d^6 + 40*a^2*b^4*c^6*d^6 - 128*a^4*c^8*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x - (b^9*c^3*d^6 + 12*a
*b^7*c^4*d^6 - 264*a^2*b^5*c^5*d^6 + 1280*a^3*b^3*c^6*d^6 - 1920*a^4*b*c^7*d^6)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^
2*c^5))/(c*x^2 + b*x + a)^(3/2)