3.1114 \(\int (A+B x) (d+e x)^m (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=282 \[ \frac{(d+e x)^{m+3} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6 (m+3)}-\frac{(d+e x)^{m+4} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6 (m+4)}-\frac{d^2 (B d-A e) (c d-b e)^2 (d+e x)^{m+1}}{e^6 (m+1)}+\frac{d (c d-b e) (d+e x)^{m+2} (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (m+2)}-\frac{c (d+e x)^{m+5} (-A c e-2 b B e+5 B c d)}{e^6 (m+5)}+\frac{B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]

[Out]

-((d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A
*e*(2*c*d - b*e))*(d + e*x)^(2 + m))/(e^6*(2 + m)) + ((A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2
- 12*b*c*d*e + 3*b^2*e^2))*(d + e*x)^(3 + m))/(e^6*(3 + m)) - ((2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c
*d*e + b^2*e^2))*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(5 + m))/(e^6*(5
+ m)) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6 + m))

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Rubi [A]  time = 0.227059, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{(d+e x)^{m+3} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6 (m+3)}-\frac{(d+e x)^{m+4} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6 (m+4)}-\frac{d^2 (B d-A e) (c d-b e)^2 (d+e x)^{m+1}}{e^6 (m+1)}+\frac{d (c d-b e) (d+e x)^{m+2} (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (m+2)}-\frac{c (d+e x)^{m+5} (-A c e-2 b B e+5 B c d)}{e^6 (m+5)}+\frac{B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + (d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A
*e*(2*c*d - b*e))*(d + e*x)^(2 + m))/(e^6*(2 + m)) + ((A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2
- 12*b*c*d*e + 3*b^2*e^2))*(d + e*x)^(3 + m))/(e^6*(3 + m)) - ((2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c
*d*e + b^2*e^2))*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(5 + m))/(e^6*(5
+ m)) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6 + m))

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^m \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac{d^2 (B d-A e) (c d-b e)^2 (d+e x)^m}{e^5}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{1+m}}{e^5}+\frac{\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{2+m}}{e^5}+\frac{\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{3+m}}{e^5}+\frac{c (-5 B c d+2 b B e+A c e) (d+e x)^{4+m}}{e^5}+\frac{B c^2 (d+e x)^{5+m}}{e^5}\right ) \, dx\\ &=-\frac{d^2 (B d-A e) (c d-b e)^2 (d+e x)^{1+m}}{e^6 (1+m)}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{2+m}}{e^6 (2+m)}+\frac{\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{3+m}}{e^6 (3+m)}-\frac{\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{4+m}}{e^6 (4+m)}-\frac{c (5 B c d-2 b B e-A c e) (d+e x)^{5+m}}{e^6 (5+m)}+\frac{B c^2 (d+e x)^{6+m}}{e^6 (6+m)}\\ \end{align*}

Mathematica [A]  time = 0.468557, size = 309, normalized size = 1.1 \[ \frac{(d+e x)^{m+1} \left (A e \left (\frac{(d+e x)^2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{m+3}+\frac{d^2 (c d-b e)^2}{m+1}-\frac{2 c (d+e x)^3 (2 c d-b e)}{m+4}-\frac{2 d (d+e x) (c d-b e) (2 c d-b e)}{m+2}+\frac{c^2 (d+e x)^4}{m+5}\right )+B \left (\frac{(d+e x)^3 \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{m+4}-\frac{d (d+e x)^2 \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{m+3}+\frac{d^2 (d+e x) (5 c d-3 b e) (c d-b e)}{m+2}-\frac{d^3 (c d-b e)^2}{m+1}-\frac{c (d+e x)^4 (5 c d-2 b e)}{m+5}+\frac{c^2 (d+e x)^5}{m+6}\right )\right )}{e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(A*e*((d^2*(c*d - b*e)^2)/(1 + m) - (2*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x))/(2 + m) + ((6
*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^2)/(3 + m) - (2*c*(2*c*d - b*e)*(d + e*x)^3)/(4 + m) + (c^2*(d + e*x
)^4)/(5 + m)) + B*(-((d^3*(c*d - b*e)^2)/(1 + m)) + (d^2*(5*c*d - 3*b*e)*(c*d - b*e)*(d + e*x))/(2 + m) - (d*(
10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)*(d + e*x)^2)/(3 + m) + ((10*c^2*d^2 - 8*b*c*d*e + b^2*e^2)*(d + e*x)^3)/(
4 + m) - (c*(5*c*d - 2*b*e)*(d + e*x)^4)/(5 + m) + (c^2*(d + e*x)^5)/(6 + m))))/e^6

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Maple [B]  time = 0.01, size = 1616, 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)^m*(c*x^2+b*x)^2,x)

[Out]

(e*x+d)^(1+m)*(B*c^2*e^5*m^5*x^5+A*c^2*e^5*m^5*x^4+2*B*b*c*e^5*m^5*x^4+15*B*c^2*e^5*m^4*x^5+2*A*b*c*e^5*m^5*x^
3+16*A*c^2*e^5*m^4*x^4+B*b^2*e^5*m^5*x^3+32*B*b*c*e^5*m^4*x^4-5*B*c^2*d*e^4*m^4*x^4+85*B*c^2*e^5*m^3*x^5+A*b^2
*e^5*m^5*x^2+34*A*b*c*e^5*m^4*x^3-4*A*c^2*d*e^4*m^4*x^3+95*A*c^2*e^5*m^3*x^4+17*B*b^2*e^5*m^4*x^3-8*B*b*c*d*e^
4*m^4*x^3+190*B*b*c*e^5*m^3*x^4-50*B*c^2*d*e^4*m^3*x^4+225*B*c^2*e^5*m^2*x^5+18*A*b^2*e^5*m^4*x^2-6*A*b*c*d*e^
4*m^4*x^2+214*A*b*c*e^5*m^3*x^3-48*A*c^2*d*e^4*m^3*x^3+260*A*c^2*e^5*m^2*x^4-3*B*b^2*d*e^4*m^4*x^2+107*B*b^2*e
^5*m^3*x^3-96*B*b*c*d*e^4*m^3*x^3+520*B*b*c*e^5*m^2*x^4+20*B*c^2*d^2*e^3*m^3*x^3-175*B*c^2*d*e^4*m^2*x^4+274*B
*c^2*e^5*m*x^5-2*A*b^2*d*e^4*m^4*x+121*A*b^2*e^5*m^3*x^2-84*A*b*c*d*e^4*m^3*x^2+614*A*b*c*e^5*m^2*x^3+12*A*c^2
*d^2*e^3*m^3*x^2-188*A*c^2*d*e^4*m^2*x^3+324*A*c^2*e^5*m*x^4-42*B*b^2*d*e^4*m^3*x^2+307*B*b^2*e^5*m^2*x^3+24*B
*b*c*d^2*e^3*m^3*x^2-376*B*b*c*d*e^4*m^2*x^3+648*B*b*c*e^5*m*x^4+120*B*c^2*d^2*e^3*m^2*x^3-250*B*c^2*d*e^4*m*x
^4+120*B*c^2*e^5*x^5-32*A*b^2*d*e^4*m^3*x+372*A*b^2*e^5*m^2*x^2+12*A*b*c*d^2*e^3*m^3*x-390*A*b*c*d*e^4*m^2*x^2
+792*A*b*c*e^5*m*x^3+108*A*c^2*d^2*e^3*m^2*x^2-288*A*c^2*d*e^4*m*x^3+144*A*c^2*e^5*x^4+6*B*b^2*d^2*e^3*m^3*x-1
95*B*b^2*d*e^4*m^2*x^2+396*B*b^2*e^5*m*x^3+216*B*b*c*d^2*e^3*m^2*x^2-576*B*b*c*d*e^4*m*x^3+288*B*b*c*e^5*x^4-6
0*B*c^2*d^3*e^2*m^2*x^2+220*B*c^2*d^2*e^3*m*x^3-120*B*c^2*d*e^4*x^4+2*A*b^2*d^2*e^3*m^3-178*A*b^2*d*e^4*m^2*x+
508*A*b^2*e^5*m*x^2+144*A*b*c*d^2*e^3*m^2*x-672*A*b*c*d*e^4*m*x^2+360*A*b*c*e^5*x^3-24*A*c^2*d^3*e^2*m^2*x+240
*A*c^2*d^2*e^3*m*x^2-144*A*c^2*d*e^4*x^3+72*B*b^2*d^2*e^3*m^2*x-336*B*b^2*d*e^4*m*x^2+180*B*b^2*e^5*x^3-48*B*b
*c*d^3*e^2*m^2*x+480*B*b*c*d^2*e^3*m*x^2-288*B*b*c*d*e^4*x^3-180*B*c^2*d^3*e^2*m*x^2+120*B*c^2*d^2*e^3*x^3+30*
A*b^2*d^2*e^3*m^2-388*A*b^2*d*e^4*m*x+240*A*b^2*e^5*x^2-12*A*b*c*d^3*e^2*m^2+492*A*b*c*d^2*e^3*m*x-360*A*b*c*d
*e^4*x^2-168*A*c^2*d^3*e^2*m*x+144*A*c^2*d^2*e^3*x^2-6*B*b^2*d^3*e^2*m^2+246*B*b^2*d^2*e^3*m*x-180*B*b^2*d*e^4
*x^2-336*B*b*c*d^3*e^2*m*x+288*B*b*c*d^2*e^3*x^2+120*B*c^2*d^4*e*m*x-120*B*c^2*d^3*e^2*x^2+148*A*b^2*d^2*e^3*m
-240*A*b^2*d*e^4*x-132*A*b*c*d^3*e^2*m+360*A*b*c*d^2*e^3*x+24*A*c^2*d^4*e*m-144*A*c^2*d^3*e^2*x-66*B*b^2*d^3*e
^2*m+180*B*b^2*d^2*e^3*x+48*B*b*c*d^4*e*m-288*B*b*c*d^3*e^2*x+120*B*c^2*d^4*e*x+240*A*b^2*d^2*e^3-360*A*b*c*d^
3*e^2+144*A*c^2*d^4*e-180*B*b^2*d^3*e^2+288*B*b*c*d^4*e-120*B*c^2*d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^
2+1764*m+720)

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Maxima [B]  time = 1.15365, size = 1019, normalized size = 3.61 \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)^m*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*A*b^2/((m^3 + 6*m^2 + 11*m +
6)*e^3) + ((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3
*e*m*x - 6*d^4)*(e*x + d)^m*B*b^2/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 2*((m^3 + 6*m^2 + 11*m + 6)*e^4*
x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*A*b*c/((m^4 +
10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 2*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 +
6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x +
d)^m*B*b*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^
5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24
*d^4*e*m*x + 24*d^5)*(e*x + d)^m*A*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + ((m^5 + 15*m^4
+ 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^4 + 6*m^3
+ 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x -
120*d^6)*(e*x + d)^m*B*c^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6)

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Fricas [B]  time = 1.73505, size = 3054, normalized size = 10.83 \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)^m*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

(2*A*b^2*d^3*e^3*m^3 - 120*B*c^2*d^6 + 240*A*b^2*d^3*e^3 + 144*(2*B*b*c + A*c^2)*d^5*e - 180*(B*b^2 + 2*A*b*c)
*d^4*e^2 + (B*c^2*e^6*m^5 + 15*B*c^2*e^6*m^4 + 85*B*c^2*e^6*m^3 + 225*B*c^2*e^6*m^2 + 274*B*c^2*e^6*m + 120*B*
c^2*e^6)*x^6 + (144*(2*B*b*c + A*c^2)*e^6 + (B*c^2*d*e^5 + (2*B*b*c + A*c^2)*e^6)*m^5 + 2*(5*B*c^2*d*e^5 + 8*(
2*B*b*c + A*c^2)*e^6)*m^4 + 5*(7*B*c^2*d*e^5 + 19*(2*B*b*c + A*c^2)*e^6)*m^3 + 10*(5*B*c^2*d*e^5 + 26*(2*B*b*c
+ A*c^2)*e^6)*m^2 + 12*(2*B*c^2*d*e^5 + 27*(2*B*b*c + A*c^2)*e^6)*m)*x^5 + (180*(B*b^2 + 2*A*b*c)*e^6 + ((2*B
*b*c + A*c^2)*d*e^5 + (B*b^2 + 2*A*b*c)*e^6)*m^5 - (5*B*c^2*d^2*e^4 - 12*(2*B*b*c + A*c^2)*d*e^5 - 17*(B*b^2 +
2*A*b*c)*e^6)*m^4 - (30*B*c^2*d^2*e^4 - 47*(2*B*b*c + A*c^2)*d*e^5 - 107*(B*b^2 + 2*A*b*c)*e^6)*m^3 - (55*B*c
^2*d^2*e^4 - 72*(2*B*b*c + A*c^2)*d*e^5 - 307*(B*b^2 + 2*A*b*c)*e^6)*m^2 - 6*(5*B*c^2*d^2*e^4 - 6*(2*B*b*c + A
*c^2)*d*e^5 - 66*(B*b^2 + 2*A*b*c)*e^6)*m)*x^4 + (240*A*b^2*e^6 + (A*b^2*e^6 + (B*b^2 + 2*A*b*c)*d*e^5)*m^5 +
2*(9*A*b^2*e^6 - 2*(2*B*b*c + A*c^2)*d^2*e^4 + 7*(B*b^2 + 2*A*b*c)*d*e^5)*m^4 + (20*B*c^2*d^3*e^3 + 121*A*b^2*
e^6 - 36*(2*B*b*c + A*c^2)*d^2*e^4 + 65*(B*b^2 + 2*A*b*c)*d*e^5)*m^3 + 4*(15*B*c^2*d^3*e^3 + 93*A*b^2*e^6 - 20
*(2*B*b*c + A*c^2)*d^2*e^4 + 28*(B*b^2 + 2*A*b*c)*d*e^5)*m^2 + 4*(10*B*c^2*d^3*e^3 + 127*A*b^2*e^6 - 12*(2*B*b
*c + A*c^2)*d^2*e^4 + 15*(B*b^2 + 2*A*b*c)*d*e^5)*m)*x^3 + 6*(5*A*b^2*d^3*e^3 - (B*b^2 + 2*A*b*c)*d^4*e^2)*m^2
+ (A*b^2*d*e^5*m^5 + (16*A*b^2*d*e^5 - 3*(B*b^2 + 2*A*b*c)*d^2*e^4)*m^4 + (89*A*b^2*d*e^5 + 12*(2*B*b*c + A*c
^2)*d^3*e^3 - 36*(B*b^2 + 2*A*b*c)*d^2*e^4)*m^3 - (60*B*c^2*d^4*e^2 - 194*A*b^2*d*e^5 - 84*(2*B*b*c + A*c^2)*d
^3*e^3 + 123*(B*b^2 + 2*A*b*c)*d^2*e^4)*m^2 - 6*(10*B*c^2*d^4*e^2 - 20*A*b^2*d*e^5 - 12*(2*B*b*c + A*c^2)*d^3*
e^3 + 15*(B*b^2 + 2*A*b*c)*d^2*e^4)*m)*x^2 + 2*(74*A*b^2*d^3*e^3 + 12*(2*B*b*c + A*c^2)*d^5*e - 33*(B*b^2 + 2*
A*b*c)*d^4*e^2)*m - 2*(A*b^2*d^2*e^4*m^4 + 3*(5*A*b^2*d^2*e^4 - (B*b^2 + 2*A*b*c)*d^3*e^3)*m^3 + (74*A*b^2*d^2
*e^4 + 12*(2*B*b*c + A*c^2)*d^4*e^2 - 33*(B*b^2 + 2*A*b*c)*d^3*e^3)*m^2 - 6*(10*B*c^2*d^5*e - 20*A*b^2*d^2*e^4
- 12*(2*B*b*c + A*c^2)*d^4*e^2 + 15*(B*b^2 + 2*A*b*c)*d^3*e^3)*m)*x)*(e*x + d)^m/(e^6*m^6 + 21*e^6*m^5 + 175*
e^6*m^4 + 735*e^6*m^3 + 1624*e^6*m^2 + 1764*e^6*m + 720*e^6)

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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)**m*(c*x**2+b*x)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.27747, size = 3816, normalized size = 13.53 \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)^m*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

((x*e + d)^m*B*c^2*m^5*x^6*e^6 + (x*e + d)^m*B*c^2*d*m^5*x^5*e^5 + 2*(x*e + d)^m*B*b*c*m^5*x^5*e^6 + (x*e + d)
^m*A*c^2*m^5*x^5*e^6 + 15*(x*e + d)^m*B*c^2*m^4*x^6*e^6 + 2*(x*e + d)^m*B*b*c*d*m^5*x^4*e^5 + (x*e + d)^m*A*c^
2*d*m^5*x^4*e^5 + 10*(x*e + d)^m*B*c^2*d*m^4*x^5*e^5 - 5*(x*e + d)^m*B*c^2*d^2*m^4*x^4*e^4 + (x*e + d)^m*B*b^2
*m^5*x^4*e^6 + 2*(x*e + d)^m*A*b*c*m^5*x^4*e^6 + 32*(x*e + d)^m*B*b*c*m^4*x^5*e^6 + 16*(x*e + d)^m*A*c^2*m^4*x
^5*e^6 + 85*(x*e + d)^m*B*c^2*m^3*x^6*e^6 + (x*e + d)^m*B*b^2*d*m^5*x^3*e^5 + 2*(x*e + d)^m*A*b*c*d*m^5*x^3*e^
5 + 24*(x*e + d)^m*B*b*c*d*m^4*x^4*e^5 + 12*(x*e + d)^m*A*c^2*d*m^4*x^4*e^5 + 35*(x*e + d)^m*B*c^2*d*m^3*x^5*e
^5 - 8*(x*e + d)^m*B*b*c*d^2*m^4*x^3*e^4 - 4*(x*e + d)^m*A*c^2*d^2*m^4*x^3*e^4 - 30*(x*e + d)^m*B*c^2*d^2*m^3*
x^4*e^4 + 20*(x*e + d)^m*B*c^2*d^3*m^3*x^3*e^3 + (x*e + d)^m*A*b^2*m^5*x^3*e^6 + 17*(x*e + d)^m*B*b^2*m^4*x^4*
e^6 + 34*(x*e + d)^m*A*b*c*m^4*x^4*e^6 + 190*(x*e + d)^m*B*b*c*m^3*x^5*e^6 + 95*(x*e + d)^m*A*c^2*m^3*x^5*e^6
+ 225*(x*e + d)^m*B*c^2*m^2*x^6*e^6 + (x*e + d)^m*A*b^2*d*m^5*x^2*e^5 + 14*(x*e + d)^m*B*b^2*d*m^4*x^3*e^5 + 2
8*(x*e + d)^m*A*b*c*d*m^4*x^3*e^5 + 94*(x*e + d)^m*B*b*c*d*m^3*x^4*e^5 + 47*(x*e + d)^m*A*c^2*d*m^3*x^4*e^5 +
50*(x*e + d)^m*B*c^2*d*m^2*x^5*e^5 - 3*(x*e + d)^m*B*b^2*d^2*m^4*x^2*e^4 - 6*(x*e + d)^m*A*b*c*d^2*m^4*x^2*e^4
- 72*(x*e + d)^m*B*b*c*d^2*m^3*x^3*e^4 - 36*(x*e + d)^m*A*c^2*d^2*m^3*x^3*e^4 - 55*(x*e + d)^m*B*c^2*d^2*m^2*
x^4*e^4 + 24*(x*e + d)^m*B*b*c*d^3*m^3*x^2*e^3 + 12*(x*e + d)^m*A*c^2*d^3*m^3*x^2*e^3 + 60*(x*e + d)^m*B*c^2*d
^3*m^2*x^3*e^3 - 60*(x*e + d)^m*B*c^2*d^4*m^2*x^2*e^2 + 18*(x*e + d)^m*A*b^2*m^4*x^3*e^6 + 107*(x*e + d)^m*B*b
^2*m^3*x^4*e^6 + 214*(x*e + d)^m*A*b*c*m^3*x^4*e^6 + 520*(x*e + d)^m*B*b*c*m^2*x^5*e^6 + 260*(x*e + d)^m*A*c^2
*m^2*x^5*e^6 + 274*(x*e + d)^m*B*c^2*m*x^6*e^6 + 16*(x*e + d)^m*A*b^2*d*m^4*x^2*e^5 + 65*(x*e + d)^m*B*b^2*d*m
^3*x^3*e^5 + 130*(x*e + d)^m*A*b*c*d*m^3*x^3*e^5 + 144*(x*e + d)^m*B*b*c*d*m^2*x^4*e^5 + 72*(x*e + d)^m*A*c^2*
d*m^2*x^4*e^5 + 24*(x*e + d)^m*B*c^2*d*m*x^5*e^5 - 2*(x*e + d)^m*A*b^2*d^2*m^4*x*e^4 - 36*(x*e + d)^m*B*b^2*d^
2*m^3*x^2*e^4 - 72*(x*e + d)^m*A*b*c*d^2*m^3*x^2*e^4 - 160*(x*e + d)^m*B*b*c*d^2*m^2*x^3*e^4 - 80*(x*e + d)^m*
A*c^2*d^2*m^2*x^3*e^4 - 30*(x*e + d)^m*B*c^2*d^2*m*x^4*e^4 + 6*(x*e + d)^m*B*b^2*d^3*m^3*x*e^3 + 12*(x*e + d)^
m*A*b*c*d^3*m^3*x*e^3 + 168*(x*e + d)^m*B*b*c*d^3*m^2*x^2*e^3 + 84*(x*e + d)^m*A*c^2*d^3*m^2*x^2*e^3 + 40*(x*e
+ d)^m*B*c^2*d^3*m*x^3*e^3 - 48*(x*e + d)^m*B*b*c*d^4*m^2*x*e^2 - 24*(x*e + d)^m*A*c^2*d^4*m^2*x*e^2 - 60*(x*
e + d)^m*B*c^2*d^4*m*x^2*e^2 + 120*(x*e + d)^m*B*c^2*d^5*m*x*e + 121*(x*e + d)^m*A*b^2*m^3*x^3*e^6 + 307*(x*e
+ d)^m*B*b^2*m^2*x^4*e^6 + 614*(x*e + d)^m*A*b*c*m^2*x^4*e^6 + 648*(x*e + d)^m*B*b*c*m*x^5*e^6 + 324*(x*e + d)
^m*A*c^2*m*x^5*e^6 + 120*(x*e + d)^m*B*c^2*x^6*e^6 + 89*(x*e + d)^m*A*b^2*d*m^3*x^2*e^5 + 112*(x*e + d)^m*B*b^
2*d*m^2*x^3*e^5 + 224*(x*e + d)^m*A*b*c*d*m^2*x^3*e^5 + 72*(x*e + d)^m*B*b*c*d*m*x^4*e^5 + 36*(x*e + d)^m*A*c^
2*d*m*x^4*e^5 - 30*(x*e + d)^m*A*b^2*d^2*m^3*x*e^4 - 123*(x*e + d)^m*B*b^2*d^2*m^2*x^2*e^4 - 246*(x*e + d)^m*A
*b*c*d^2*m^2*x^2*e^4 - 96*(x*e + d)^m*B*b*c*d^2*m*x^3*e^4 - 48*(x*e + d)^m*A*c^2*d^2*m*x^3*e^4 + 2*(x*e + d)^m
*A*b^2*d^3*m^3*e^3 + 66*(x*e + d)^m*B*b^2*d^3*m^2*x*e^3 + 132*(x*e + d)^m*A*b*c*d^3*m^2*x*e^3 + 144*(x*e + d)^
m*B*b*c*d^3*m*x^2*e^3 + 72*(x*e + d)^m*A*c^2*d^3*m*x^2*e^3 - 6*(x*e + d)^m*B*b^2*d^4*m^2*e^2 - 12*(x*e + d)^m*
A*b*c*d^4*m^2*e^2 - 288*(x*e + d)^m*B*b*c*d^4*m*x*e^2 - 144*(x*e + d)^m*A*c^2*d^4*m*x*e^2 + 48*(x*e + d)^m*B*b
*c*d^5*m*e + 24*(x*e + d)^m*A*c^2*d^5*m*e - 120*(x*e + d)^m*B*c^2*d^6 + 372*(x*e + d)^m*A*b^2*m^2*x^3*e^6 + 39
6*(x*e + d)^m*B*b^2*m*x^4*e^6 + 792*(x*e + d)^m*A*b*c*m*x^4*e^6 + 288*(x*e + d)^m*B*b*c*x^5*e^6 + 144*(x*e + d
)^m*A*c^2*x^5*e^6 + 194*(x*e + d)^m*A*b^2*d*m^2*x^2*e^5 + 60*(x*e + d)^m*B*b^2*d*m*x^3*e^5 + 120*(x*e + d)^m*A
*b*c*d*m*x^3*e^5 - 148*(x*e + d)^m*A*b^2*d^2*m^2*x*e^4 - 90*(x*e + d)^m*B*b^2*d^2*m*x^2*e^4 - 180*(x*e + d)^m*
A*b*c*d^2*m*x^2*e^4 + 30*(x*e + d)^m*A*b^2*d^3*m^2*e^3 + 180*(x*e + d)^m*B*b^2*d^3*m*x*e^3 + 360*(x*e + d)^m*A
*b*c*d^3*m*x*e^3 - 66*(x*e + d)^m*B*b^2*d^4*m*e^2 - 132*(x*e + d)^m*A*b*c*d^4*m*e^2 + 288*(x*e + d)^m*B*b*c*d^
5*e + 144*(x*e + d)^m*A*c^2*d^5*e + 508*(x*e + d)^m*A*b^2*m*x^3*e^6 + 180*(x*e + d)^m*B*b^2*x^4*e^6 + 360*(x*e
+ d)^m*A*b*c*x^4*e^6 + 120*(x*e + d)^m*A*b^2*d*m*x^2*e^5 - 240*(x*e + d)^m*A*b^2*d^2*m*x*e^4 + 148*(x*e + d)^
m*A*b^2*d^3*m*e^3 - 180*(x*e + d)^m*B*b^2*d^4*e^2 - 360*(x*e + d)^m*A*b*c*d^4*e^2 + 240*(x*e + d)^m*A*b^2*x^3*
e^6 + 240*(x*e + d)^m*A*b^2*d^3*e^3)/(m^6*e^6 + 21*m^5*e^6 + 175*m^4*e^6 + 735*m^3*e^6 + 1624*m^2*e^6 + 1764*m
*e^6 + 720*e^6)