3.2490 \(\int \frac{(A+B x) (d+e x)^3}{(a+b x+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=264 \[ -\frac{16 (-2 a e+x (2 c d-b e)+b d) \left (-8 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (a A e^2+3 a B d e+4 A c d^2\right )+b^2 e (3 A e+5 B d)\right )}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^3 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{4 (d+e x)^2 \left (-x \left (4 c (3 a B e+4 A c d)-8 b c (A e+B d)+b^2 B e\right )-8 b (a B e+A c d)+4 a A c e+b^2 (3 A e+4 B d)\right )}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^3)/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) - (4*(d + e*x)^2*(4
*a*A*c*e + b^2*(4*B*d + 3*A*e) - 8*b*(A*c*d + a*B*e) - (b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + 3*a*B*e))
*x))/(15*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) - (16*(b^2*e*(5*B*d + 3*A*e) + 4*c*(4*A*c*d^2 + 3*a*B*d*e +
a*A*e^2) - 8*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*(b*d - 2*a*e + (2*c*d - b*e)*x))/(15*(b^2 - 4*a*c)^3*Sqrt[a +
b*x + c*x^2])

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Rubi [A]  time = 0.361626, antiderivative size = 264, 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, 804, 636} \[ -\frac{16 (-2 a e+x (2 c d-b e)+b d) \left (-8 b \left (a B e^2+2 A c d e+B c d^2\right )+4 c \left (a A e^2+3 a B d e+4 A c d^2\right )+b^2 e (3 A e+5 B d)\right )}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^3 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{4 (d+e x)^2 \left (-x \left (4 c (3 a B e+4 A c d)-8 b c (A e+B d)+b^2 B e\right )-8 b (a B e+A c d)+4 a A c e+b^2 (3 A e+4 B d)\right )}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^3)/(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)^3)/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) - (4*(d + e*x)^2*(4
*a*A*c*e + b^2*(4*B*d + 3*A*e) - 8*b*(A*c*d + a*B*e) - (b^2*B*e - 8*b*c*(B*d + A*e) + 4*c*(4*A*c*d + 3*a*B*e))
*x))/(15*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) - (16*(b^2*e*(5*B*d + 3*A*e) + 4*c*(4*A*c*d^2 + 3*a*B*d*e +
a*A*e^2) - 8*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*(b*d - 2*a*e + (2*c*d - b*e)*x))/(15*(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 804

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)*(b*f - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(m
*(b*(e*f + d*g) - 2*(c*d*f + a*e*g)))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)
, 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] && EqQ[Sim
plify[m + 2*p + 3], 0] && LtQ[p, -1]

Rule 636

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

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^3}{\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)^3}{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 b B d+8 A c d-3 A b e+6 a B e-(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)^3}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{4 (d+e x)^2 \left (4 a A c e+b^2 (4 B d+3 A e)-8 b (A c d+a B e)-\left (b^2 B e-8 b c (B d+A e)+4 c (4 A c d+3 a B e)\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{\left (8 \left (b^2 e (5 B d+3 A e)+4 c \left (4 A c d^2+3 a B d e+a A e^2\right )-8 b \left (B c d^2+2 A c d e+a B e^2\right )\right )\right ) \int \frac{d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 \left (b^2-4 a c\right )^2}\\ &=-\frac{2 (A b-2 a B-(b B-2 A c) x) (d+e x)^3}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac{4 (d+e x)^2 \left (4 a A c e+b^2 (4 B d+3 A e)-8 b (A c d+a B e)-\left (b^2 B e-8 b c (B d+A e)+4 c (4 A c d+3 a B e)\right ) x\right )}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac{16 \left (b^2 e (5 B d+3 A e)+4 c \left (4 A c d^2+3 a B d e+a A e^2\right )-8 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) (b d-2 a e+(2 c d-b e) x)}{15 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [B]  time = 1.8226, size = 963, normalized size = 3.65 \[ \frac{2 \left (A \left (-3 \left (d^3+5 e x d^2+15 e^2 x^2 d-5 e^3 x^3\right ) b^5+\left (10 c x \left (d^3+12 e x d^2-27 e^2 x^2 d+2 e^3 x^3\right )-6 a e \left (d^2+10 e x d-15 e^2 x^2\right )\right ) b^4-8 \left (3 a^2 (d-5 e x) e^2+c^2 x^2 \left (10 d^3-90 e x d^2+45 e^2 x^2 d-e^3 x^3\right )-5 a c \left (d^3+9 e x d^2-15 e^2 x^2 d+5 e^3 x^3\right )\right ) b^3+48 \left (a^3 e^3+a^2 c \left (3 d^2-15 e x d+5 e^2 x^2\right ) e+c^3 d x^3 \left (-10 d^2+20 e x d-3 e^2 x^2\right )+5 a c^2 x \left (-d^3+6 e x d^2-3 e^2 x^2 d+e^3 x^3\right )\right ) b^2+16 c \left (8 c^3 d^2 (3 e x-5 d) x^4+6 a c^2 \left (-10 d^3+10 e x d^2-5 e^2 x^2 d+e^3 x^3\right ) x^2-15 a^2 c (d-e x)^3+2 a^3 e^2 (5 e x-9 d)\right ) b-32 c \left (8 c^4 d^3 x^5+2 a c^3 d \left (10 d^2+3 e^2 x^2\right ) x^3+15 a^2 c^2 d \left (d^2+e^2 x^2\right ) x-2 a^4 e^3-a^3 c e \left (9 d^2+5 e^2 x^2\right )\right )\right )+B \left (64 e^2 (3 c d-2 b e) a^4+16 \left (6 \left (d^3+5 e^2 x^2 d\right ) c^2-2 b e \left (9 d^2-15 e x d+10 e^2 x^2\right ) c+b^2 e^2 (9 d-20 e x)\right ) a^3-24 \left (e \left (d^2-15 e x d+10 e^2 x^2\right ) b^3-2 c \left (d^3-15 e x d^2+15 e^2 x^2 d-10 e^3 x^3\right ) b^2+10 c^2 x (e x-d)^3 b+4 c^3 e x^3 \left (5 d^2+e^2 x^2\right )\right ) a^2-2 \left (96 c^4 d^2 e x^5-16 b c^3 d \left (10 d^2-15 e x d+9 e^2 x^2\right ) x^3+24 b^2 c^2 \left (-10 d^3+15 e x d^2-15 e^2 x^2 d+e^3 x^3\right ) x^2+60 b^3 c \left (-d^3+5 e x d^2-5 e^2 x^2 d+e^3 x^3\right ) x+b^4 \left (d^3+30 e x d^2-135 e^2 x^2 d+20 e^3 x^3\right )\right ) a+b x \left (-5 \left (d^3+9 e x d^2-9 e^2 x^2 d-e^3 x^3\right ) b^4+2 c x \left (20 d^3-135 e x d^2+30 e^2 x^2 d+e^3 x^3\right ) b^3+24 c^2 d x^2 \left (10 d^2-15 e x d+e^2 x^2\right ) b^2+16 c^3 d^2 x^3 (20 d-9 e x) b+128 c^4 d^3 x^4\right )\right )\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*(A*(-3*b^5*(d^3 + 5*d^2*e*x + 15*d*e^2*x^2 - 5*e^3*x^3) - 32*c*(-2*a^4*e^3 + 8*c^4*d^3*x^5 + 15*a^2*c^2*d*x
*(d^2 + e^2*x^2) + 2*a*c^3*d*x^3*(10*d^2 + 3*e^2*x^2) - a^3*c*e*(9*d^2 + 5*e^2*x^2)) + 16*b*c*(-15*a^2*c*(d -
e*x)^3 + 8*c^3*d^2*x^4*(-5*d + 3*e*x) + 2*a^3*e^2*(-9*d + 5*e*x) + 6*a*c^2*x^2*(-10*d^3 + 10*d^2*e*x - 5*d*e^2
*x^2 + e^3*x^3)) + 48*b^2*(a^3*e^3 + c^3*d*x^3*(-10*d^2 + 20*d*e*x - 3*e^2*x^2) + a^2*c*e*(3*d^2 - 15*d*e*x +
5*e^2*x^2) + 5*a*c^2*x*(-d^3 + 6*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + b^4*(-6*a*e*(d^2 + 10*d*e*x - 15*e^2*x^2)
+ 10*c*x*(d^3 + 12*d^2*e*x - 27*d*e^2*x^2 + 2*e^3*x^3)) - 8*b^3*(3*a^2*e^2*(d - 5*e*x) + c^2*x^2*(10*d^3 - 90
*d^2*e*x + 45*d*e^2*x^2 - e^3*x^3) - 5*a*c*(d^3 + 9*d^2*e*x - 15*d*e^2*x^2 + 5*e^3*x^3))) + B*(64*a^4*e^2*(3*c
*d - 2*b*e) + 16*a^3*(b^2*e^2*(9*d - 20*e*x) - 2*b*c*e*(9*d^2 - 15*d*e*x + 10*e^2*x^2) + 6*c^2*(d^3 + 5*d*e^2*
x^2)) - 24*a^2*(10*b*c^2*x*(-d + e*x)^3 + 4*c^3*e*x^3*(5*d^2 + e^2*x^2) + b^3*e*(d^2 - 15*d*e*x + 10*e^2*x^2)
- 2*b^2*c*(d^3 - 15*d^2*e*x + 15*d*e^2*x^2 - 10*e^3*x^3)) + b*x*(128*c^4*d^3*x^4 + 16*b*c^3*d^2*x^3*(20*d - 9*
e*x) + 24*b^2*c^2*d*x^2*(10*d^2 - 15*d*e*x + e^2*x^2) - 5*b^4*(d^3 + 9*d^2*e*x - 9*d*e^2*x^2 - e^3*x^3) + 2*b^
3*c*x*(20*d^3 - 135*d^2*e*x + 30*d*e^2*x^2 + e^3*x^3)) - 2*a*(96*c^4*d^2*e*x^5 - 16*b*c^3*d*x^3*(10*d^2 - 15*d
*e*x + 9*e^2*x^2) + 24*b^2*c^2*x^2*(-10*d^3 + 15*d^2*e*x - 15*d*e^2*x^2 + e^3*x^3) + 60*b^3*c*x*(-d^3 + 5*d^2*
e*x - 5*d*e^2*x^2 + e^3*x^3) + b^4*(d^3 + 30*d^2*e*x - 135*d*e^2*x^2 + 20*e^3*x^3)))))/(15*(b^2 - 4*a*c)^3*(a
+ x*(b + c*x))^(5/2))

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Maple [B]  time = 0.013, size = 1502, normalized size = 5.7 \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)^3/(c*x^2+b*x+a)^(7/2),x)

[Out]

-2/15/(c*x^2+b*x+a)^(5/2)*(96*A*a*b*c^3*e^3*x^5-192*A*a*c^4*d*e^2*x^5+8*A*b^3*c^2*e^3*x^5-144*A*b^2*c^3*d*e^2*
x^5+384*A*b*c^4*d^2*e*x^5-256*A*c^5*d^3*x^5-96*B*a^2*c^3*e^3*x^5-48*B*a*b^2*c^2*e^3*x^5+288*B*a*b*c^3*d*e^2*x^
5-192*B*a*c^4*d^2*e*x^5+2*B*b^4*c*e^3*x^5+24*B*b^3*c^2*d*e^2*x^5-144*B*b^2*c^3*d^2*e*x^5+128*B*b*c^4*d^3*x^5+2
40*A*a*b^2*c^2*e^3*x^4-480*A*a*b*c^3*d*e^2*x^4+20*A*b^4*c*e^3*x^4-360*A*b^3*c^2*d*e^2*x^4+960*A*b^2*c^3*d^2*e*
x^4-640*A*b*c^4*d^3*x^4-240*B*a^2*b*c^2*e^3*x^4-120*B*a*b^3*c*e^3*x^4+720*B*a*b^2*c^2*d*e^2*x^4-480*B*a*b*c^3*
d^2*e*x^4+5*B*b^5*e^3*x^4+60*B*b^4*c*d*e^2*x^4-360*B*b^3*c^2*d^2*e*x^4+320*B*b^2*c^3*d^3*x^4+240*A*a^2*b*c^2*e
^3*x^3-480*A*a^2*c^3*d*e^2*x^3+200*A*a*b^3*c*e^3*x^3-720*A*a*b^2*c^2*d*e^2*x^3+960*A*a*b*c^3*d^2*e*x^3-640*A*a
*c^4*d^3*x^3+15*A*b^5*e^3*x^3-270*A*b^4*c*d*e^2*x^3+720*A*b^3*c^2*d^2*e*x^3-480*A*b^2*c^3*d^3*x^3-480*B*a^2*b^
2*c*e^3*x^3+720*B*a^2*b*c^2*d*e^2*x^3-480*B*a^2*c^3*d^2*e*x^3-40*B*a*b^4*e^3*x^3+600*B*a*b^3*c*d*e^2*x^3-720*B
*a*b^2*c^2*d^2*e*x^3+320*B*a*b*c^3*d^3*x^3+45*B*b^5*d*e^2*x^3-270*B*b^4*c*d^2*e*x^3+240*B*b^3*c^2*d^3*x^3+160*
A*a^3*c^2*e^3*x^2+240*A*a^2*b^2*c*e^3*x^2-720*A*a^2*b*c^2*d*e^2*x^2+90*A*a*b^4*e^3*x^2-600*A*a*b^3*c*d*e^2*x^2
+1440*A*a*b^2*c^2*d^2*e*x^2-960*A*a*b*c^3*d^3*x^2-45*A*b^5*d*e^2*x^2+120*A*b^4*c*d^2*e*x^2-80*A*b^3*c^2*d^3*x^
2-320*B*a^3*b*c*e^3*x^2+480*B*a^3*c^2*d*e^2*x^2-240*B*a^2*b^3*e^3*x^2+720*B*a^2*b^2*c*d*e^2*x^2-720*B*a^2*b*c^
2*d^2*e*x^2+270*B*a*b^4*d*e^2*x^2-600*B*a*b^3*c*d^2*e*x^2+480*B*a*b^2*c^2*d^3*x^2-45*B*b^5*d^2*e*x^2+40*B*b^4*
c*d^3*x^2+160*A*a^3*b*c*e^3*x+120*A*a^2*b^3*e^3*x-720*A*a^2*b^2*c*d*e^2*x+720*A*a^2*b*c^2*d^2*e*x-480*A*a^2*c^
3*d^3*x-60*A*a*b^4*d*e^2*x+360*A*a*b^3*c*d^2*e*x-240*A*a*b^2*c^2*d^3*x-15*A*b^5*d^2*e*x+10*A*b^4*c*d^3*x-320*B
*a^3*b^2*e^3*x+480*B*a^3*b*c*d*e^2*x+360*B*a^2*b^3*d*e^2*x-720*B*a^2*b^2*c*d^2*e*x+240*B*a^2*b*c^2*d^3*x-60*B*
a*b^4*d^2*e*x+120*B*a*b^3*c*d^3*x-5*B*b^5*d^3*x+64*A*a^4*c*e^3+48*A*a^3*b^2*e^3-288*A*a^3*b*c*d*e^2+288*A*a^3*
c^2*d^2*e-24*A*a^2*b^3*d*e^2+144*A*a^2*b^2*c*d^2*e-240*A*a^2*b*c^2*d^3-6*A*a*b^4*d^2*e+40*A*a*b^3*c*d^3-3*A*b^
5*d^3-128*B*a^4*b*e^3+192*B*a^4*c*d*e^2+144*B*a^3*b^2*d*e^2-288*B*a^3*b*c*d^2*e+96*B*a^3*c^2*d^3-24*B*a^2*b^3*
d^2*e+48*B*a^2*b^2*c*d^3-2*B*a*b^4*d^3)/(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)^3/(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)^3/(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)**3/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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Giac [B]  time = 1.1967, size = 1989, normalized size = 7.53 \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)^3/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")

[Out]

1/15*(((((2*(64*B*b*c^4*d^3 - 128*A*c^5*d^3 - 72*B*b^2*c^3*d^2*e - 96*B*a*c^4*d^2*e + 192*A*b*c^4*d^2*e + 12*B
*b^3*c^2*d*e^2 + 144*B*a*b*c^3*d*e^2 - 72*A*b^2*c^3*d*e^2 - 96*A*a*c^4*d*e^2 + B*b^4*c*e^3 - 24*B*a*b^2*c^2*e^
3 + 4*A*b^3*c^2*e^3 - 48*B*a^2*c^3*e^3 + 48*A*a*b*c^3*e^3)*x/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3
*c^6) + 5*(64*B*b^2*c^3*d^3 - 128*A*b*c^4*d^3 - 72*B*b^3*c^2*d^2*e - 96*B*a*b*c^3*d^2*e + 192*A*b^2*c^3*d^2*e
+ 12*B*b^4*c*d*e^2 + 144*B*a*b^2*c^2*d*e^2 - 72*A*b^3*c^2*d*e^2 - 96*A*a*b*c^3*d*e^2 + B*b^5*e^3 - 24*B*a*b^3*
c*e^3 + 4*A*b^4*c*e^3 - 48*B*a^2*b*c^2*e^3 + 48*A*a*b^2*c^2*e^3)/(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^3 + 64*B*a*b*c^3*d^3 - 96*A*b^2*c^3*d^3 - 128*A*a*c^4*d^3 - 54*B*b^4*c*d^2*e
- 144*B*a*b^2*c^2*d^2*e + 144*A*b^3*c^2*d^2*e - 96*B*a^2*c^3*d^2*e + 192*A*a*b*c^3*d^2*e + 9*B*b^5*d*e^2 + 120
*B*a*b^3*c*d*e^2 - 54*A*b^4*c*d*e^2 + 144*B*a^2*b*c^2*d*e^2 - 144*A*a*b^2*c^2*d*e^2 - 96*A*a^2*c^3*d*e^2 - 8*B
*a*b^4*e^3 + 3*A*b^5*e^3 - 96*B*a^2*b^2*c*e^3 + 40*A*a*b^3*c*e^3 + 48*A*a^2*b*c^2*e^3)/(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^3 + 96*B*a*b^2*c^2*d^3 - 16*A*b^3*c^2*d^3 - 192*A*a*b*c^3*
d^3 - 9*B*b^5*d^2*e - 120*B*a*b^3*c*d^2*e + 24*A*b^4*c*d^2*e - 144*B*a^2*b*c^2*d^2*e + 288*A*a*b^2*c^2*d^2*e +
54*B*a*b^4*d*e^2 - 9*A*b^5*d*e^2 + 144*B*a^2*b^2*c*d*e^2 - 120*A*a*b^3*c*d*e^2 + 96*B*a^3*c^2*d*e^2 - 144*A*a
^2*b*c^2*d*e^2 - 48*B*a^2*b^3*e^3 + 18*A*a*b^4*e^3 - 64*B*a^3*b*c*e^3 + 48*A*a^2*b^2*c*e^3 + 32*A*a^3*c^2*e^3)
/(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^3 - 24*B*a*b^3*c*d^3 - 2*A*b^4*c*d^3 -
48*B*a^2*b*c^2*d^3 + 48*A*a*b^2*c^2*d^3 + 96*A*a^2*c^3*d^3 + 12*B*a*b^4*d^2*e + 3*A*b^5*d^2*e + 144*B*a^2*b^2
*c*d^2*e - 72*A*a*b^3*c*d^2*e - 144*A*a^2*b*c^2*d^2*e - 72*B*a^2*b^3*d*e^2 + 12*A*a*b^4*d*e^2 - 96*B*a^3*b*c*d
*e^2 + 144*A*a^2*b^2*c*d*e^2 + 64*B*a^3*b^2*e^3 - 24*A*a^2*b^3*e^3 - 32*A*a^3*b*c*e^3)/(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^3 + 3*A*b^5*d^3 - 48*B*a^2*b^2*c*d^3 - 40*A*a*b^3*c*d^3 - 96
*B*a^3*c^2*d^3 + 240*A*a^2*b*c^2*d^3 + 24*B*a^2*b^3*d^2*e + 6*A*a*b^4*d^2*e + 288*B*a^3*b*c*d^2*e - 144*A*a^2*
b^2*c*d^2*e - 288*A*a^3*c^2*d^2*e - 144*B*a^3*b^2*d*e^2 + 24*A*a^2*b^3*d*e^2 - 192*B*a^4*c*d*e^2 + 288*A*a^3*b
*c*d*e^2 + 128*B*a^4*b*e^3 - 48*A*a^3*b^2*e^3 - 64*A*a^4*c*e^3)/(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)