3.2491 \(\int \frac{(A+B x) (d+e x)^2}{(a+b x+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=324 \[ \frac{8 (b+2 c x) \left (4 b c \left (3 a B e^2+8 A c d e+4 B c d^2\right )-8 c^2 \left (a A e^2+2 a B d e+4 A c d^2\right )-6 b^2 c e (A e+2 B d)+b^3 B e^2\right )}{15 c \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{8 \left (x \left (-4 c^2 \left (-a A e^2+a B d e+2 A c d^2\right )-3 b^2 c e (A e+B d)+4 b c^2 d (2 A e+B d)+b^3 B e^2\right )+b^2 \left (a B e^2+A c d e+2 B c d^2\right )-4 b c \left (a A e^2+2 a B d e+A c d^2\right )+4 a c e (a B e+3 A c d)\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]

[Out]

(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^2)/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) - (8*(4*a*c*e*(3*A*
c*d + a*B*e) - 4*b*c*(A*c*d^2 + 2*a*B*d*e + a*A*e^2) + b^2*(2*B*c*d^2 + A*c*d*e + a*B*e^2) + (b^3*B*e^2 - 3*b^
2*c*e*(B*d + A*e) + 4*b*c^2*d*(B*d + 2*A*e) - 4*c^2*(2*A*c*d^2 + a*B*d*e - a*A*e^2))*x))/(15*c*(b^2 - 4*a*c)^2
*(a + b*x + c*x^2)^(3/2)) + (8*(b^3*B*e^2 - 6*b^2*c*e*(2*B*d + A*e) - 8*c^2*(4*A*c*d^2 + 2*a*B*d*e + a*A*e^2)
+ 4*b*c*(4*B*c*d^2 + 8*A*c*d*e + 3*a*B*e^2))*(b + 2*c*x))/(15*c*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.385266, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {820, 777, 613} \[ \frac{8 (b+2 c x) \left (4 b c \left (3 a B e^2+8 A c d e+4 B c d^2\right )-8 c^2 \left (a A e^2+2 a B d e+4 A c d^2\right )-6 b^2 c e (A e+2 B d)+b^3 B e^2\right )}{15 c \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{8 \left (x \left (-4 c^2 \left (-a A e^2+a B d e+2 A c d^2\right )-3 b^2 c e (A e+B d)+4 b c^2 d (2 A e+B d)+b^3 B e^2\right )+b^2 \left (a B e^2+A c d e+2 B c d^2\right )-4 b c \left (a A e^2+2 a B d e+A c d^2\right )+4 a c e (a B e+3 A c d)\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^2)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^2)/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) - (8*(4*a*c*e*(3*A*
c*d + a*B*e) - 4*b*c*(A*c*d^2 + 2*a*B*d*e + a*A*e^2) + b^2*(2*B*c*d^2 + A*c*d*e + a*B*e^2) + (b^3*B*e^2 - 3*b^
2*c*e*(B*d + A*e) + 4*b*c^2*d*(B*d + 2*A*e) - 4*c^2*(2*A*c*d^2 + a*B*d*e - a*A*e^2))*x))/(15*c*(b^2 - 4*a*c)^2
*(a + b*x + c*x^2)^(3/2)) + (8*(b^3*B*e^2 - 6*b^2*c*e*(2*B*d + A*e) - 8*c^2*(4*A*c*d^2 + 2*a*B*d*e + a*A*e^2)
+ 4*b*c*(4*B*c*d^2 + 8*A*c*d*e + 3*a*B*e^2))*(b + 2*c*x))/(15*c*(b^2 - 4*a*c)^3*Sqrt[a + b*x + c*x^2])

Rule 820

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

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
 + c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=-\frac{2 (A b-2 a B-(b B-2 A c) x) (d+e x)^2}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{2 \int \frac{(d+e x) (2 (4 A c d+2 a B e-b (2 B d+A e))-2 (b B-2 A c) e x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 \left (b^2-4 a c\right )}\\ &=-\frac{2 (A b-2 a B-(b B-2 A c) x) (d+e x)^2}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{8 \left (4 a c e (3 A c d+a B e)-4 b c \left (A c d^2+2 a B d e+a A e^2\right )+b^2 \left (2 B c d^2+A c d e+a B e^2\right )+\left (b^3 B e^2-3 b^2 c e (B d+A e)+4 b c^2 d (B d+2 A e)-4 c^2 \left (2 A c d^2+a B d e-a A e^2\right )\right ) x\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{\left (4 \left (b^3 B e^2-6 b^2 c e (2 B d+A e)-8 c^2 \left (4 A c d^2+2 a B d e+a A e^2\right )+4 b c \left (4 B c d^2+8 A c d e+3 a B e^2\right )\right )\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 c \left (b^2-4 a c\right )^2}\\ &=-\frac{2 (A b-2 a B-(b B-2 A c) x) (d+e x)^2}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{8 \left (4 a c e (3 A c d+a B e)-4 b c \left (A c d^2+2 a B d e+a A e^2\right )+b^2 \left (2 B c d^2+A c d e+a B e^2\right )+\left (b^3 B e^2-3 b^2 c e (B d+A e)+4 b c^2 d (B d+2 A e)-4 c^2 \left (2 A c d^2+a B d e-a A e^2\right )\right ) x\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{8 \left (b^3 B e^2-6 b^2 c e (2 B d+A e)-8 c^2 \left (4 A c d^2+2 a B d e+a A e^2\right )+4 b c \left (4 B c d^2+8 A c d e+3 a B e^2\right )\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [B]  time = 1.61415, size = 711, normalized size = 2.19 \[ \frac{2 B \left (16 a^3 \left (3 b^2 e^2+2 b c e (5 e x-6 d)+2 c^2 \left (3 d^2+5 e^2 x^2\right )\right )-8 a^2 \left (-6 b^2 c \left (d^2-10 d e x+5 e^2 x^2\right )+b^3 e (2 d-15 e x)-30 b c^2 x (d-e x)^2+40 c^3 d e x^3\right )+64 a^4 c e^2-2 a \left (-120 b^2 c^2 x^2 \left (2 d^2-2 d e x+e^2 x^2\right )-20 b^3 c x \left (3 d^2-10 d e x+5 e^2 x^2\right )+b^4 \left (d^2+20 d e x-45 e^2 x^2\right )-16 b c^3 x^3 \left (10 d^2-10 d e x+3 e^2 x^2\right )+64 c^4 d e x^5\right )+b x \left (8 b^2 c^2 x^2 \left (30 d^2-30 d e x+e^2 x^2\right )+20 b^3 c x \left (2 d^2-9 d e x+e^2 x^2\right )-5 b^4 \left (d^2+6 d e x-3 e^2 x^2\right )+32 b c^3 d x^3 (10 d-3 e x)+128 c^4 d^2 x^4\right )\right )-2 A \left (8 b^3 \left (a^2 e^2-5 a c \left (d^2+6 d e x-5 e^2 x^2\right )+5 c^2 x^2 \left (2 d^2-12 d e x+3 e^2 x^2\right )\right )+16 b^2 c \left (3 a^2 e (5 e x-2 d)+15 a c x \left (d^2-4 d e x+e^2 x^2\right )+c^2 x^3 \left (30 d^2-40 d e x+3 e^2 x^2\right )\right )+16 b c \left (15 a^2 c (d-e x)^2+6 a^3 e^2+10 a c^2 x^2 \left (6 d^2-4 d e x+e^2 x^2\right )+8 c^3 d x^4 (5 d-2 e x)\right )+32 c^2 \left (5 a^2 c x \left (3 d^2+e^2 x^2\right )-6 a^3 d e+2 a c^2 x^3 \left (10 d^2+e^2 x^2\right )+8 c^3 d^2 x^5\right )+2 b^4 \left (2 a e (d+5 e x)-5 c x \left (d^2+8 d e x-9 e^2 x^2\right )\right )+b^5 \left (3 d^2+10 d e x+15 e^2 x^2\right )\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^2)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(-2*A*(b^5*(3*d^2 + 10*d*e*x + 15*e^2*x^2) + 2*b^4*(2*a*e*(d + 5*e*x) - 5*c*x*(d^2 + 8*d*e*x - 9*e^2*x^2)) + 3
2*c^2*(-6*a^3*d*e + 8*c^3*d^2*x^5 + 5*a^2*c*x*(3*d^2 + e^2*x^2) + 2*a*c^2*x^3*(10*d^2 + e^2*x^2)) + 16*b*c*(6*
a^3*e^2 + 8*c^3*d*x^4*(5*d - 2*e*x) + 15*a^2*c*(d - e*x)^2 + 10*a*c^2*x^2*(6*d^2 - 4*d*e*x + e^2*x^2)) + 16*b^
2*c*(3*a^2*e*(-2*d + 5*e*x) + 15*a*c*x*(d^2 - 4*d*e*x + e^2*x^2) + c^2*x^3*(30*d^2 - 40*d*e*x + 3*e^2*x^2)) +
8*b^3*(a^2*e^2 - 5*a*c*(d^2 + 6*d*e*x - 5*e^2*x^2) + 5*c^2*x^2*(2*d^2 - 12*d*e*x + 3*e^2*x^2))) + 2*B*(64*a^4*
c*e^2 + b*x*(128*c^4*d^2*x^4 + 32*b*c^3*d*x^3*(10*d - 3*e*x) - 5*b^4*(d^2 + 6*d*e*x - 3*e^2*x^2) + 8*b^2*c^2*x
^2*(30*d^2 - 30*d*e*x + e^2*x^2) + 20*b^3*c*x*(2*d^2 - 9*d*e*x + e^2*x^2)) + 16*a^3*(3*b^2*e^2 + 2*b*c*e*(-6*d
 + 5*e*x) + 2*c^2*(3*d^2 + 5*e^2*x^2)) - 8*a^2*(40*c^3*d*e*x^3 + b^3*e*(2*d - 15*e*x) - 30*b*c^2*x*(d - e*x)^2
 - 6*b^2*c*(d^2 - 10*d*e*x + 5*e^2*x^2)) - 2*a*(64*c^4*d*e*x^5 + b^4*(d^2 + 20*d*e*x - 45*e^2*x^2) - 120*b^2*c
^2*x^2*(2*d^2 - 2*d*e*x + e^2*x^2) - 16*b*c^3*x^3*(10*d^2 - 10*d*e*x + 3*e^2*x^2) - 20*b^3*c*x*(3*d^2 - 10*d*e
*x + 5*e^2*x^2))))/(15*(b^2 - 4*a*c)^3*(a + x*(b + c*x))^(5/2))

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Maple [B]  time = 0.012, size = 1064, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^2/(c*x^2+b*x+a)^(7/2),x)

[Out]

2/15/(c*x^2+b*x+a)^(5/2)*(64*A*a*c^4*e^2*x^5+48*A*b^2*c^3*e^2*x^5-256*A*b*c^4*d*e*x^5+256*A*c^5*d^2*x^5-96*B*a
*b*c^3*e^2*x^5+128*B*a*c^4*d*e*x^5-8*B*b^3*c^2*e^2*x^5+96*B*b^2*c^3*d*e*x^5-128*B*b*c^4*d^2*x^5+160*A*a*b*c^3*
e^2*x^4+120*A*b^3*c^2*e^2*x^4-640*A*b^2*c^3*d*e*x^4+640*A*b*c^4*d^2*x^4-240*B*a*b^2*c^2*e^2*x^4+320*B*a*b*c^3*
d*e*x^4-20*B*b^4*c*e^2*x^4+240*B*b^3*c^2*d*e*x^4-320*B*b^2*c^3*d^2*x^4+160*A*a^2*c^3*e^2*x^3+240*A*a*b^2*c^2*e
^2*x^3-640*A*a*b*c^3*d*e*x^3+640*A*a*c^4*d^2*x^3+90*A*b^4*c*e^2*x^3-480*A*b^3*c^2*d*e*x^3+480*A*b^2*c^3*d^2*x^
3-240*B*a^2*b*c^2*e^2*x^3+320*B*a^2*c^3*d*e*x^3-200*B*a*b^3*c*e^2*x^3+480*B*a*b^2*c^2*d*e*x^3-320*B*a*b*c^3*d^
2*x^3-15*B*b^5*e^2*x^3+180*B*b^4*c*d*e*x^3-240*B*b^3*c^2*d^2*x^3+240*A*a^2*b*c^2*e^2*x^2+200*A*a*b^3*c*e^2*x^2
-960*A*a*b^2*c^2*d*e*x^2+960*A*a*b*c^3*d^2*x^2+15*A*b^5*e^2*x^2-80*A*b^4*c*d*e*x^2+80*A*b^3*c^2*d^2*x^2-160*B*
a^3*c^2*e^2*x^2-240*B*a^2*b^2*c*e^2*x^2+480*B*a^2*b*c^2*d*e*x^2-90*B*a*b^4*e^2*x^2+400*B*a*b^3*c*d*e*x^2-480*B
*a*b^2*c^2*d^2*x^2+30*B*b^5*d*e*x^2-40*B*b^4*c*d^2*x^2+240*A*a^2*b^2*c*e^2*x-480*A*a^2*b*c^2*d*e*x+480*A*a^2*c
^3*d^2*x+20*A*a*b^4*e^2*x-240*A*a*b^3*c*d*e*x+240*A*a*b^2*c^2*d^2*x+10*A*b^5*d*e*x-10*A*b^4*c*d^2*x-160*B*a^3*
b*c*e^2*x-120*B*a^2*b^3*e^2*x+480*B*a^2*b^2*c*d*e*x-240*B*a^2*b*c^2*d^2*x+40*B*a*b^4*d*e*x-120*B*a*b^3*c*d^2*x
+5*B*b^5*d^2*x+96*A*a^3*b*c*e^2-192*A*a^3*c^2*d*e+8*A*a^2*b^3*e^2-96*A*a^2*b^2*c*d*e+240*A*a^2*b*c^2*d^2+4*A*a
*b^4*d*e-40*A*a*b^3*c*d^2+3*A*b^5*d^2-64*B*a^4*c*e^2-48*B*a^3*b^2*e^2+192*B*a^3*b*c*d*e-96*B*a^3*c^2*d^2+16*B*
a^2*b^3*d*e-48*B*a^2*b^2*c*d^2+2*B*a*b^4*d^2)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)

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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((B*x+A)*(e*x+d)^2/(c*x^2+b*x+a)^(7/2),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((B*x+A)*(e*x+d)^2/(c*x^2+b*x+a)^(7/2),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((B*x+A)*(e*x+d)**2/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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Giac [B]  time = 1.18755, size = 1519, normalized size = 4.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^2/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")

[Out]

1/15*((((4*(2*(16*B*b*c^4*d^2 - 32*A*c^5*d^2 - 12*B*b^2*c^3*d*e - 16*B*a*c^4*d*e + 32*A*b*c^4*d*e + B*b^3*c^2*
e^2 + 12*B*a*b*c^3*e^2 - 6*A*b^2*c^3*e^2 - 8*A*a*c^4*e^2)*x/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*
c^6) + 5*(16*B*b^2*c^3*d^2 - 32*A*b*c^4*d^2 - 12*B*b^3*c^2*d*e - 16*B*a*b*c^3*d*e + 32*A*b^2*c^3*d*e + B*b^4*c
*e^2 + 12*B*a*b^2*c^2*e^2 - 6*A*b^3*c^2*e^2 - 8*A*a*b*c^3*e^2)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a
^3*c^6))*x + 5*(48*B*b^3*c^2*d^2 + 64*B*a*b*c^3*d^2 - 96*A*b^2*c^3*d^2 - 128*A*a*c^4*d^2 - 36*B*b^4*c*d*e - 96
*B*a*b^2*c^2*d*e + 96*A*b^3*c^2*d*e - 64*B*a^2*c^3*d*e + 128*A*a*b*c^3*d*e + 3*B*b^5*e^2 + 40*B*a*b^3*c*e^2 -
18*A*b^4*c*e^2 + 48*B*a^2*b*c^2*e^2 - 48*A*a*b^2*c^2*e^2 - 32*A*a^2*c^3*e^2)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*
b^2*c^5 - 64*a^3*c^6))*x + 5*(8*B*b^4*c*d^2 + 96*B*a*b^2*c^2*d^2 - 16*A*b^3*c^2*d^2 - 192*A*a*b*c^3*d^2 - 6*B*
b^5*d*e - 80*B*a*b^3*c*d*e + 16*A*b^4*c*d*e - 96*B*a^2*b*c^2*d*e + 192*A*a*b^2*c^2*d*e + 18*B*a*b^4*e^2 - 3*A*
b^5*e^2 + 48*B*a^2*b^2*c*e^2 - 40*A*a*b^3*c*e^2 + 32*B*a^3*c^2*e^2 - 48*A*a^2*b*c^2*e^2)/(b^6*c^3 - 12*a*b^4*c
^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x - 5*(B*b^5*d^2 - 24*B*a*b^3*c*d^2 - 2*A*b^4*c*d^2 - 48*B*a^2*b*c^2*d^2 +
48*A*a*b^2*c^2*d^2 + 96*A*a^2*c^3*d^2 + 8*B*a*b^4*d*e + 2*A*b^5*d*e + 96*B*a^2*b^2*c*d*e - 48*A*a*b^3*c*d*e -
96*A*a^2*b*c^2*d*e - 24*B*a^2*b^3*e^2 + 4*A*a*b^4*e^2 - 32*B*a^3*b*c*e^2 + 48*A*a^2*b^2*c*e^2)/(b^6*c^3 - 12*a
*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x - (2*B*a*b^4*d^2 + 3*A*b^5*d^2 - 48*B*a^2*b^2*c*d^2 - 40*A*a*b^3*c*
d^2 - 96*B*a^3*c^2*d^2 + 240*A*a^2*b*c^2*d^2 + 16*B*a^2*b^3*d*e + 4*A*a*b^4*d*e + 192*B*a^3*b*c*d*e - 96*A*a^2
*b^2*c*d*e - 192*A*a^3*c^2*d*e - 48*B*a^3*b^2*e^2 + 8*A*a^2*b^3*e^2 - 64*B*a^4*c*e^2 + 96*A*a^3*b*c*e^2)/(b^6*
c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))/(c*x^2 + b*x + a)^(5/2)