3.1127 \(\int (A+B x) (d+e x)^m (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=484 \[ \frac{3 d (c d-b e) (d+e x)^{m+3} \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8 (m+3)}+\frac{(d+e x)^{m+4} \left (B d \left (30 b^2 c d e^2-4 b^3 e^3-60 b c^2 d^2 e+35 c^3 d^3\right )-A e \left (12 b^2 c d e^2-b^3 e^3-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^8 (m+4)}+\frac{(d+e x)^{m+5} \left (3 A c e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B \left (15 b^2 c d e^2-b^3 e^3-45 b c^2 d^2 e+35 c^3 d^3\right )\right )}{e^8 (m+5)}-\frac{3 c (d+e x)^{m+6} \left (A c e (2 c d-b e)-B \left (b^2 e^2-6 b c d e+7 c^2 d^2\right )\right )}{e^8 (m+6)}-\frac{c^2 (d+e x)^{m+7} (-A c e-3 b B e+7 B c d)}{e^8 (m+7)}-\frac{d^3 (B d-A e) (c d-b e)^3 (d+e x)^{m+1}}{e^8 (m+1)}+\frac{d^2 (c d-b e)^2 (d+e x)^{m+2} (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8 (m+2)}+\frac{B c^3 (d+e x)^{m+8}}{e^8 (m+8)} \]

[Out]

-((d^3*(B*d - A*e)*(c*d - b*e)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + (d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) -
3*A*e*(2*c*d - b*e))*(d + e*x)^(2 + m))/(e^8*(2 + m)) + (3*d*(c*d - b*e)*(A*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^
2) - B*d*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2))*(d + e*x)^(3 + m))/(e^8*(3 + m)) + ((B*d*(35*c^3*d^3 - 60*b*c^2*
d^2*e + 30*b^2*c*d*e^2 - 4*b^3*e^3) - A*e*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3))*(d + e*x)^
(4 + m))/(e^8*(4 + m)) + ((3*A*c*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2) - B*(35*c^3*d^3 - 45*b*c^2*d^2*e + 15*b^2
*c*d*e^2 - b^3*e^3))*(d + e*x)^(5 + m))/(e^8*(5 + m)) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 - 6*b*c*d*e +
b^2*e^2))*(d + e*x)^(6 + m))/(e^8*(6 + m)) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)
) + (B*c^3*(d + e*x)^(8 + m))/(e^8*(8 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.407169, antiderivative size = 484, 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} \[ \frac{3 d (c d-b e) (d+e x)^{m+3} \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8 (m+3)}+\frac{(d+e x)^{m+4} \left (B d \left (30 b^2 c d e^2-4 b^3 e^3-60 b c^2 d^2 e+35 c^3 d^3\right )-A e \left (12 b^2 c d e^2-b^3 e^3-30 b c^2 d^2 e+20 c^3 d^3\right )\right )}{e^8 (m+4)}+\frac{(d+e x)^{m+5} \left (3 A c e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B \left (15 b^2 c d e^2-b^3 e^3-45 b c^2 d^2 e+35 c^3 d^3\right )\right )}{e^8 (m+5)}-\frac{3 c (d+e x)^{m+6} \left (A c e (2 c d-b e)-B \left (b^2 e^2-6 b c d e+7 c^2 d^2\right )\right )}{e^8 (m+6)}-\frac{c^2 (d+e x)^{m+7} (-A c e-3 b B e+7 B c d)}{e^8 (m+7)}-\frac{d^3 (B d-A e) (c d-b e)^3 (d+e x)^{m+1}}{e^8 (m+1)}+\frac{d^2 (c d-b e)^2 (d+e x)^{m+2} (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8 (m+2)}+\frac{B c^3 (d+e x)^{m+8}}{e^8 (m+8)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^m*(b*x + c*x^2)^3,x]

[Out]

-((d^3*(B*d - A*e)*(c*d - b*e)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + (d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) -
3*A*e*(2*c*d - b*e))*(d + e*x)^(2 + m))/(e^8*(2 + m)) + (3*d*(c*d - b*e)*(A*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^
2) - B*d*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2))*(d + e*x)^(3 + m))/(e^8*(3 + m)) + ((B*d*(35*c^3*d^3 - 60*b*c^2*
d^2*e + 30*b^2*c*d*e^2 - 4*b^3*e^3) - A*e*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3))*(d + e*x)^
(4 + m))/(e^8*(4 + m)) + ((3*A*c*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2) - B*(35*c^3*d^3 - 45*b*c^2*d^2*e + 15*b^2
*c*d*e^2 - b^3*e^3))*(d + e*x)^(5 + m))/(e^8*(5 + m)) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 - 6*b*c*d*e +
b^2*e^2))*(d + e*x)^(6 + m))/(e^8*(6 + m)) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)
) + (B*c^3*(d + e*x)^(8 + m))/(e^8*(8 + 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 )^3 \, dx &=\int \left (-\frac{d^3 (B d-A e) (c d-b e)^3 (d+e x)^m}{e^7}+\frac{d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e)) (d+e x)^{1+m}}{e^7}+\frac{3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right ) (d+e x)^{2+m}}{e^7}+\frac{\left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{3+m}}{e^7}+\frac{\left (3 A c e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B \left (35 c^3 d^3-45 b c^2 d^2 e+15 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{4+m}}{e^7}+\frac{3 c \left (-A c e (2 c d-b e)+B \left (7 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) (d+e x)^{5+m}}{e^7}+\frac{c^2 (-7 B c d+3 b B e+A c e) (d+e x)^{6+m}}{e^7}+\frac{B c^3 (d+e x)^{7+m}}{e^7}\right ) \, dx\\ &=-\frac{d^3 (B d-A e) (c d-b e)^3 (d+e x)^{1+m}}{e^8 (1+m)}+\frac{d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e)) (d+e x)^{2+m}}{e^8 (2+m)}+\frac{3 d (c d-b e) \left (A e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B d \left (7 c^2 d^2-8 b c d e+2 b^2 e^2\right )\right ) (d+e x)^{3+m}}{e^8 (3+m)}+\frac{\left (B d \left (35 c^3 d^3-60 b c^2 d^2 e+30 b^2 c d e^2-4 b^3 e^3\right )-A e \left (20 c^3 d^3-30 b c^2 d^2 e+12 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{4+m}}{e^8 (4+m)}+\frac{\left (3 A c e \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )-B \left (35 c^3 d^3-45 b c^2 d^2 e+15 b^2 c d e^2-b^3 e^3\right )\right ) (d+e x)^{5+m}}{e^8 (5+m)}-\frac{3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) (d+e x)^{6+m}}{e^8 (6+m)}-\frac{c^2 (7 B c d-3 b B e-A c e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac{B c^3 (d+e x)^{8+m}}{e^8 (8+m)}\\ \end{align*}

Mathematica [A]  time = 1.03366, size = 525, normalized size = 1.08 \[ \frac{(d+e x)^{m+1} \left (A e \left (\frac{3 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{m+5}-\frac{(d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{m+4}+\frac{3 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{m+3}-\frac{3 c^2 (d+e x)^5 (2 c d-b e)}{m+6}-\frac{3 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)}{m+2}+\frac{d^3 (c d-b e)^3}{m+1}+\frac{c^3 (d+e x)^6}{m+7}\right )+B \left (\frac{3 c (d+e x)^5 \left (b^2 e^2-6 b c d e+7 c^2 d^2\right )}{m+6}-\frac{(d+e x)^4 \left (15 b^2 c d e^2-b^3 e^3-45 b c^2 d^2 e+35 c^3 d^3\right )}{m+5}+\frac{d (d+e x)^3 \left (30 b^2 c d e^2-4 b^3 e^3-60 b c^2 d^2 e+35 c^3 d^3\right )}{m+4}-\frac{3 d^2 (d+e x)^2 (c d-b e) \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )}{m+3}-\frac{c^2 (d+e x)^6 (7 c d-3 b e)}{m+7}+\frac{d^3 (d+e x) (7 c d-4 b e) (c d-b e)^2}{m+2}-\frac{d^4 (c d-b e)^3}{m+1}+\frac{c^3 (d+e x)^7}{m+8}\right )\right )}{e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(A*e*((d^3*(c*d - b*e)^3)/(1 + m) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x))/(2 + m) +
(3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2)/(3 + m) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c
*d*e + b^2*e^2)*(d + e*x)^3)/(4 + m) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4)/(5 + m) - (3*c^2*(2
*c*d - b*e)*(d + e*x)^5)/(6 + m) + (c^3*(d + e*x)^6)/(7 + m)) + B*(-((d^4*(c*d - b*e)^3)/(1 + m)) + (d^3*(7*c*
d - 4*b*e)*(c*d - b*e)^2*(d + e*x))/(2 + m) - (3*d^2*(c*d - b*e)*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2)*(d + e*x)
^2)/(3 + m) + (d*(35*c^3*d^3 - 60*b*c^2*d^2*e + 30*b^2*c*d*e^2 - 4*b^3*e^3)*(d + e*x)^3)/(4 + m) - ((35*c^3*d^
3 - 45*b*c^2*d^2*e + 15*b^2*c*d*e^2 - b^3*e^3)*(d + e*x)^4)/(5 + m) + (3*c*(7*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(
d + e*x)^5)/(6 + m) - (c^2*(7*c*d - 3*b*e)*(d + e*x)^6)/(7 + m) + (c^3*(d + e*x)^7)/(8 + m))))/e^8

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 4138, normalized size = 8.6 \begin{align*} \text{output 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)^3,x)

[Out]

-(e*x+d)^(1+m)*(-B*c^3*e^7*m^7*x^7-A*c^3*e^7*m^7*x^6-3*B*b*c^2*e^7*m^7*x^6-28*B*c^3*e^7*m^6*x^7-3*A*b*c^2*e^7*
m^7*x^5-29*A*c^3*e^7*m^6*x^6-3*B*b^2*c*e^7*m^7*x^5-87*B*b*c^2*e^7*m^6*x^6+7*B*c^3*d*e^6*m^6*x^6-322*B*c^3*e^7*
m^5*x^7-3*A*b^2*c*e^7*m^7*x^4-90*A*b*c^2*e^7*m^6*x^5+6*A*c^3*d*e^6*m^6*x^5-343*A*c^3*e^7*m^5*x^6-B*b^3*e^7*m^7
*x^4-90*B*b^2*c*e^7*m^6*x^5+18*B*b*c^2*d*e^6*m^6*x^5-1029*B*b*c^2*e^7*m^5*x^6+147*B*c^3*d*e^6*m^5*x^6-1960*B*c
^3*e^7*m^4*x^7-A*b^3*e^7*m^7*x^3-93*A*b^2*c*e^7*m^6*x^4+15*A*b*c^2*d*e^6*m^6*x^4-1098*A*b*c^2*e^7*m^5*x^5+138*
A*c^3*d*e^6*m^5*x^5-2135*A*c^3*e^7*m^4*x^6-31*B*b^3*e^7*m^6*x^4+15*B*b^2*c*d*e^6*m^6*x^4-1098*B*b^2*c*e^7*m^5*
x^5+414*B*b*c^2*d*e^6*m^5*x^5-6405*B*b*c^2*e^7*m^4*x^6-42*B*c^3*d^2*e^5*m^5*x^5+1225*B*c^3*d*e^6*m^4*x^6-6769*
B*c^3*e^7*m^3*x^7-32*A*b^3*e^7*m^6*x^3+12*A*b^2*c*d*e^6*m^6*x^3-1173*A*b^2*c*e^7*m^5*x^4+375*A*b*c^2*d*e^6*m^5
*x^4-7020*A*b*c^2*e^7*m^4*x^5-30*A*c^3*d^2*e^5*m^5*x^4+1230*A*c^3*d*e^6*m^4*x^5-7504*A*c^3*e^7*m^3*x^6+4*B*b^3
*d*e^6*m^6*x^3-391*B*b^3*e^7*m^5*x^4+375*B*b^2*c*d*e^6*m^5*x^4-7020*B*b^2*c*e^7*m^4*x^5-90*B*b*c^2*d^2*e^5*m^5
*x^4+3690*B*b*c^2*d*e^6*m^4*x^5-22512*B*b*c^2*e^7*m^3*x^6-630*B*c^3*d^2*e^5*m^4*x^5+5145*B*c^3*d*e^6*m^3*x^6-1
3132*B*c^3*e^7*m^2*x^7+3*A*b^3*d*e^6*m^6*x^2-418*A*b^3*e^7*m^5*x^3+324*A*b^2*c*d*e^6*m^5*x^3-7743*A*b^2*c*e^7*
m^4*x^4-60*A*b*c^2*d^2*e^5*m^5*x^3+3615*A*b*c^2*d*e^6*m^4*x^4-25227*A*b*c^2*e^7*m^3*x^5-540*A*c^3*d^2*e^5*m^4*
x^4+5430*A*c^3*d*e^6*m^3*x^5-14756*A*c^3*e^7*m^2*x^6+108*B*b^3*d*e^6*m^5*x^3-2581*B*b^3*e^7*m^4*x^4-60*B*b^2*c
*d^2*e^5*m^5*x^3+3615*B*b^2*c*d*e^6*m^4*x^4-25227*B*b^2*c*e^7*m^3*x^5-1620*B*b*c^2*d^2*e^5*m^4*x^4+16290*B*b*c
^2*d*e^6*m^3*x^5-44268*B*b*c^2*e^7*m^2*x^6+210*B*c^3*d^3*e^4*m^4*x^4-3570*B*c^3*d^2*e^5*m^3*x^5+11368*B*c^3*d*
e^6*m^2*x^6-13068*B*c^3*e^7*m*x^7+87*A*b^3*d*e^6*m^5*x^2-2864*A*b^3*e^7*m^4*x^3-36*A*b^2*c*d^2*e^5*m^5*x^2+339
6*A*b^2*c*d*e^6*m^4*x^3-28632*A*b^2*c*e^7*m^3*x^4-1260*A*b*c^2*d^2*e^5*m^4*x^3+17025*A*b*c^2*d*e^6*m^3*x^4-504
90*A*b*c^2*e^7*m^2*x^5+120*A*c^3*d^3*e^4*m^4*x^3-3450*A*c^3*d^2*e^5*m^3*x^4+12444*A*c^3*d*e^6*m^2*x^5-14832*A*
c^3*e^7*m*x^6-12*B*b^3*d^2*e^5*m^5*x^2+1132*B*b^3*d*e^6*m^4*x^3-9544*B*b^3*e^7*m^3*x^4-1260*B*b^2*c*d^2*e^5*m^
4*x^3+17025*B*b^2*c*d*e^6*m^3*x^4-50490*B*b^2*c*e^7*m^2*x^5+360*B*b*c^2*d^3*e^4*m^4*x^3-10350*B*b*c^2*d^2*e^5*
m^3*x^4+37332*B*b*c^2*d*e^6*m^2*x^5-44496*B*b*c^2*e^7*m*x^6+2100*B*c^3*d^3*e^4*m^3*x^4-9450*B*c^3*d^2*e^5*m^2*
x^5+12348*B*c^3*d*e^6*m*x^6-5040*B*c^3*e^7*x^7-6*A*b^3*d^2*e^5*m^5*x+993*A*b^3*d*e^6*m^4*x^2-10993*A*b^3*e^7*m
^3*x^3-864*A*b^2*c*d^2*e^5*m^4*x^2+17388*A*b^2*c*d*e^6*m^3*x^3-58692*A*b^2*c*e^7*m^2*x^4+180*A*b*c^2*d^3*e^4*m
^4*x^2-9420*A*b*c^2*d^2*e^5*m^3*x^3+41010*A*b*c^2*d*e^6*m^2*x^4-51432*A*b*c^2*e^7*m*x^5+1680*A*c^3*d^3*e^4*m^3
*x^3-9900*A*c^3*d^2*e^5*m^2*x^4+13872*A*c^3*d*e^6*m*x^5-5760*A*c^3*e^7*x^6-288*B*b^3*d^2*e^5*m^4*x^2+5796*B*b^
3*d*e^6*m^3*x^3-19564*B*b^3*e^7*m^2*x^4+180*B*b^2*c*d^3*e^4*m^4*x^2-9420*B*b^2*c*d^2*e^5*m^3*x^3+41010*B*b^2*c
*d*e^6*m^2*x^4-51432*B*b^2*c*e^7*m*x^5+5040*B*b*c^2*d^3*e^4*m^3*x^3-29700*B*b*c^2*d^2*e^5*m^2*x^4+41616*B*b*c^
2*d*e^6*m*x^5-17280*B*b*c^2*e^7*x^6-840*B*c^3*d^4*e^3*m^3*x^3+7350*B*c^3*d^3*e^4*m^2*x^4-11508*B*c^3*d^2*e^5*m
*x^5+5040*B*c^3*d*e^6*x^6-162*A*b^3*d^2*e^5*m^4*x+5613*A*b^3*d*e^6*m^3*x^2-23312*A*b^3*e^7*m^2*x^3+72*A*b^2*c*
d^3*e^4*m^4*x-7596*A*b^2*c*d^2*e^5*m^3*x^2+44976*A*b^2*c*d*e^6*m^2*x^3-60912*A*b^2*c*e^7*m*x^4+3240*A*b*c^2*d^
3*e^4*m^3*x^2-30420*A*b*c^2*d^2*e^5*m^2*x^3+47400*A*b*c^2*d*e^6*m*x^4-20160*A*b*c^2*e^7*x^5-360*A*c^3*d^4*e^3*
m^3*x^2+7080*A*c^3*d^3*e^4*m^2*x^3-12720*A*c^3*d^2*e^5*m*x^4+5760*A*c^3*d*e^6*x^5+24*B*b^3*d^3*e^4*m^4*x-2532*
B*b^3*d^2*e^5*m^3*x^2+14992*B*b^3*d*e^6*m^2*x^3-20304*B*b^3*e^7*m*x^4+3240*B*b^2*c*d^3*e^4*m^3*x^2-30420*B*b^2
*c*d^2*e^5*m^2*x^3+47400*B*b^2*c*d*e^6*m*x^4-20160*B*b^2*c*e^7*x^5-1080*B*b*c^2*d^4*e^3*m^3*x^2+21240*B*b*c^2*
d^3*e^4*m^2*x^3-38160*B*b*c^2*d^2*e^5*m*x^4+17280*B*b*c^2*d*e^6*x^5-5040*B*c^3*d^4*e^3*m^2*x^3+10500*B*c^3*d^3
*e^4*m*x^4-5040*B*c^3*d^2*e^5*x^5+6*A*b^3*d^3*e^4*m^4-1662*A*b^3*d^2*e^5*m^3*x+16140*A*b^3*d*e^6*m^2*x^2-24876
*A*b^3*e^7*m*x^3+1584*A*b^2*c*d^3*e^4*m^3*x-29376*A*b^2*c*d^2*e^5*m^2*x^2+54864*A*b^2*c*d*e^6*m*x^3-24192*A*b^
2*c*e^7*x^4-360*A*b*c^2*d^4*e^3*m^3*x+18540*A*b*c^2*d^3*e^4*m^2*x^2-42360*A*b*c^2*d^2*e^5*m*x^3+20160*A*b*c^2*
d*e^6*x^4-3960*A*c^3*d^4*e^3*m^2*x^2+11280*A*c^3*d^3*e^4*m*x^3-5760*A*c^3*d^2*e^5*x^4+528*B*b^3*d^3*e^4*m^3*x-
9792*B*b^3*d^2*e^5*m^2*x^2+18288*B*b^3*d*e^6*m*x^3-8064*B*b^3*e^7*x^4-360*B*b^2*c*d^4*e^3*m^3*x+18540*B*b^2*c*
d^3*e^4*m^2*x^2-42360*B*b^2*c*d^2*e^5*m*x^3+20160*B*b^2*c*d*e^6*x^4-11880*B*b*c^2*d^4*e^3*m^2*x^2+33840*B*b*c^
2*d^3*e^4*m*x^3-17280*B*b*c^2*d^2*e^5*x^4+2520*B*c^3*d^5*e^2*m^2*x^2-9240*B*c^3*d^4*e^3*m*x^3+5040*B*c^3*d^3*e
^4*x^4+156*A*b^3*d^3*e^4*m^3-7902*A*b^3*d^2*e^5*m^2*x+21516*A*b^3*d*e^6*m*x^2-10080*A*b^3*e^7*x^3-72*A*b^2*c*d
^4*e^3*m^3+12024*A*b^2*c*d^3*e^4*m^2*x-46800*A*b^2*c*d^2*e^5*m*x^2+24192*A*b^2*c*d*e^6*x^3-5760*A*b*c^2*d^4*e^
3*m^2*x+35640*A*b*c^2*d^3*e^4*m*x^2-20160*A*b*c^2*d^2*e^5*x^3+720*A*c^3*d^5*e^2*m^2*x-9360*A*c^3*d^4*e^3*m*x^2
+5760*A*c^3*d^3*e^4*x^3-24*B*b^3*d^4*e^3*m^3+4008*B*b^3*d^3*e^4*m^2*x-15600*B*b^3*d^2*e^5*m*x^2+8064*B*b^3*d*e
^6*x^3-5760*B*b^2*c*d^4*e^3*m^2*x+35640*B*b^2*c*d^3*e^4*m*x^2-20160*B*b^2*c*d^2*e^5*x^3+2160*B*b*c^2*d^5*e^2*m
^2*x-28080*B*b*c^2*d^4*e^3*m*x^2+17280*B*b*c^2*d^3*e^4*x^3+7560*B*c^3*d^5*e^2*m*x^2-5040*B*c^3*d^4*e^3*x^3+150
6*A*b^3*d^3*e^4*m^2-16476*A*b^3*d^2*e^5*m*x+10080*A*b^3*d*e^6*x^2-1512*A*b^2*c*d^4*e^3*m^2+34704*A*b^2*c*d^3*e
^4*m*x-24192*A*b^2*c*d^2*e^5*x^2+360*A*b*c^2*d^5*e^2*m^2-25560*A*b*c^2*d^4*e^3*m*x+20160*A*b*c^2*d^3*e^4*x^2+6
480*A*c^3*d^5*e^2*m*x-5760*A*c^3*d^4*e^3*x^2-504*B*b^3*d^4*e^3*m^2+11568*B*b^3*d^3*e^4*m*x-8064*B*b^3*d^2*e^5*
x^2+360*B*b^2*c*d^5*e^2*m^2-25560*B*b^2*c*d^4*e^3*m*x+20160*B*b^2*c*d^3*e^4*x^2+19440*B*b*c^2*d^5*e^2*m*x-1728
0*B*b*c^2*d^4*e^3*x^2-5040*B*c^3*d^6*e*m*x+5040*B*c^3*d^5*e^2*x^2+6396*A*b^3*d^3*e^4*m-10080*A*b^3*d^2*e^5*x-1
0512*A*b^2*c*d^4*e^3*m+24192*A*b^2*c*d^3*e^4*x+5400*A*b*c^2*d^5*e^2*m-20160*A*b*c^2*d^4*e^3*x-720*A*c^3*d^6*e*
m+5760*A*c^3*d^5*e^2*x-3504*B*b^3*d^4*e^3*m+8064*B*b^3*d^3*e^4*x+5400*B*b^2*c*d^5*e^2*m-20160*B*b^2*c*d^4*e^3*
x-2160*B*b*c^2*d^6*e*m+17280*B*b*c^2*d^5*e^2*x-5040*B*c^3*d^6*e*x+10080*A*b^3*d^3*e^4-24192*A*b^2*c*d^4*e^3+20
160*A*b*c^2*d^5*e^2-5760*A*c^3*d^6*e-8064*B*b^3*d^4*e^3+20160*B*b^2*c*d^5*e^2-17280*B*b*c^2*d^6*e+5040*B*c^3*d
^7)/e^8/(m^8+36*m^7+546*m^6+4536*m^5+22449*m^4+67284*m^3+118124*m^2+109584*m+40320)

________________________________________________________________________________________

Maxima [B]  time = 1.22318, size = 2061, normalized size = 4.26 \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)^3,x, algorithm="maxima")

[Out]

((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^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((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^3/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + 3*((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*b^2*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274
*m + 120)*e^5) + 3*((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*b^2*c/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2
+ 1764*m + 720)*e^6) + 3*((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 + 5
0*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*A*b*c^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 162
4*m^2 + 1764*m + 720)*e^6) + 3*((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 +
15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5
+ 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4*x^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x
^2 - 720*d^6*e*m*x + 720*d^7)*(e*x + d)^m*B*b*c^2/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 +
13068*m + 5040)*e^7) + ((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5
+ 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*(
m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4*x^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 72
0*d^6*e*m*x + 720*d^7)*(e*x + d)^m*A*c^3/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m
+ 5040)*e^7) + ((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*e^8*x^8 + (m^7 + 2
1*m^6 + 175*m^5 + 735*m^4 + 1624*m^3 + 1764*m^2 + 720*m)*d*e^7*x^7 - 7*(m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*
m^2 + 120*m)*d^2*e^6*x^6 + 42*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^3*e^5*x^5 - 210*(m^4 + 6*m^3 + 11*m^2
+ 6*m)*d^4*e^4*x^4 + 840*(m^3 + 3*m^2 + 2*m)*d^5*e^3*x^3 - 2520*(m^2 + m)*d^6*e^2*x^2 + 5040*d^7*e*m*x - 5040*
d^8)*(e*x + d)^m*B*c^3/((m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 22449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 4
0320)*e^8)

________________________________________________________________________________________

Fricas [B]  time = 2.41561, size = 7102, normalized size = 14.67 \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)^3,x, algorithm="fricas")

[Out]

-(6*A*b^3*d^4*e^4*m^4 + 5040*B*c^3*d^8 + 10080*A*b^3*d^4*e^4 - 5760*(3*B*b*c^2 + A*c^3)*d^7*e + 20160*(B*b^2*c
+ A*b*c^2)*d^6*e^2 - 8064*(B*b^3 + 3*A*b^2*c)*d^5*e^3 - (B*c^3*e^8*m^7 + 28*B*c^3*e^8*m^6 + 322*B*c^3*e^8*m^5
+ 1960*B*c^3*e^8*m^4 + 6769*B*c^3*e^8*m^3 + 13132*B*c^3*e^8*m^2 + 13068*B*c^3*e^8*m + 5040*B*c^3*e^8)*x^8 - (
5760*(3*B*b*c^2 + A*c^3)*e^8 + (B*c^3*d*e^7 + (3*B*b*c^2 + A*c^3)*e^8)*m^7 + (21*B*c^3*d*e^7 + 29*(3*B*b*c^2 +
A*c^3)*e^8)*m^6 + 7*(25*B*c^3*d*e^7 + 49*(3*B*b*c^2 + A*c^3)*e^8)*m^5 + 35*(21*B*c^3*d*e^7 + 61*(3*B*b*c^2 +
A*c^3)*e^8)*m^4 + 56*(29*B*c^3*d*e^7 + 134*(3*B*b*c^2 + A*c^3)*e^8)*m^3 + 28*(63*B*c^3*d*e^7 + 527*(3*B*b*c^2
+ A*c^3)*e^8)*m^2 + 144*(5*B*c^3*d*e^7 + 103*(3*B*b*c^2 + A*c^3)*e^8)*m)*x^7 - (20160*(B*b^2*c + A*b*c^2)*e^8
+ ((3*B*b*c^2 + A*c^3)*d*e^7 + 3*(B*b^2*c + A*b*c^2)*e^8)*m^7 - (7*B*c^3*d^2*e^6 - 23*(3*B*b*c^2 + A*c^3)*d*e^
7 - 90*(B*b^2*c + A*b*c^2)*e^8)*m^6 - (105*B*c^3*d^2*e^6 - 205*(3*B*b*c^2 + A*c^3)*d*e^7 - 1098*(B*b^2*c + A*b
*c^2)*e^8)*m^5 - 5*(119*B*c^3*d^2*e^6 - 181*(3*B*b*c^2 + A*c^3)*d*e^7 - 1404*(B*b^2*c + A*b*c^2)*e^8)*m^4 - (1
575*B*c^3*d^2*e^6 - 2074*(3*B*b*c^2 + A*c^3)*d*e^7 - 25227*(B*b^2*c + A*b*c^2)*e^8)*m^3 - 2*(959*B*c^3*d^2*e^6
- 1156*(3*B*b*c^2 + A*c^3)*d*e^7 - 25245*(B*b^2*c + A*b*c^2)*e^8)*m^2 - 24*(35*B*c^3*d^2*e^6 - 40*(3*B*b*c^2
+ A*c^3)*d*e^7 - 2143*(B*b^2*c + A*b*c^2)*e^8)*m)*x^6 - (8064*(B*b^3 + 3*A*b^2*c)*e^8 + (3*(B*b^2*c + A*b*c^2)
*d*e^7 + (B*b^3 + 3*A*b^2*c)*e^8)*m^7 - (6*(3*B*b*c^2 + A*c^3)*d^2*e^6 - 75*(B*b^2*c + A*b*c^2)*d*e^7 - 31*(B*
b^3 + 3*A*b^2*c)*e^8)*m^6 + (42*B*c^3*d^3*e^5 - 108*(3*B*b*c^2 + A*c^3)*d^2*e^6 + 723*(B*b^2*c + A*b*c^2)*d*e^
7 + 391*(B*b^3 + 3*A*b^2*c)*e^8)*m^5 + (420*B*c^3*d^3*e^5 - 690*(3*B*b*c^2 + A*c^3)*d^2*e^6 + 3405*(B*b^2*c +
A*b*c^2)*d*e^7 + 2581*(B*b^3 + 3*A*b^2*c)*e^8)*m^4 + 2*(735*B*c^3*d^3*e^5 - 990*(3*B*b*c^2 + A*c^3)*d^2*e^6 +
4101*(B*b^2*c + A*b*c^2)*d*e^7 + 4772*(B*b^3 + 3*A*b^2*c)*e^8)*m^3 + 4*(525*B*c^3*d^3*e^5 - 636*(3*B*b*c^2 + A
*c^3)*d^2*e^6 + 2370*(B*b^2*c + A*b*c^2)*d*e^7 + 4891*(B*b^3 + 3*A*b^2*c)*e^8)*m^2 + 144*(7*B*c^3*d^3*e^5 - 8*
(3*B*b*c^2 + A*c^3)*d^2*e^6 + 28*(B*b^2*c + A*b*c^2)*d*e^7 + 141*(B*b^3 + 3*A*b^2*c)*e^8)*m)*x^5 - (10080*A*b^
3*e^8 + (A*b^3*e^8 + (B*b^3 + 3*A*b^2*c)*d*e^7)*m^7 + (32*A*b^3*e^8 - 15*(B*b^2*c + A*b*c^2)*d^2*e^6 + 27*(B*b
^3 + 3*A*b^2*c)*d*e^7)*m^6 + (418*A*b^3*e^8 + 30*(3*B*b*c^2 + A*c^3)*d^3*e^5 - 315*(B*b^2*c + A*b*c^2)*d^2*e^6
+ 283*(B*b^3 + 3*A*b^2*c)*d*e^7)*m^5 - (210*B*c^3*d^4*e^4 - 2864*A*b^3*e^8 - 420*(3*B*b*c^2 + A*c^3)*d^3*e^5
+ 2355*(B*b^2*c + A*b*c^2)*d^2*e^6 - 1449*(B*b^3 + 3*A*b^2*c)*d*e^7)*m^4 - (1260*B*c^3*d^4*e^4 - 10993*A*b^3*e
^8 - 1770*(3*B*b*c^2 + A*c^3)*d^3*e^5 + 7605*(B*b^2*c + A*b*c^2)*d^2*e^6 - 3748*(B*b^3 + 3*A*b^2*c)*d*e^7)*m^3
- 2*(1155*B*c^3*d^4*e^4 - 11656*A*b^3*e^8 - 1410*(3*B*b*c^2 + A*c^3)*d^3*e^5 + 5295*(B*b^2*c + A*b*c^2)*d^2*e
^6 - 2286*(B*b^3 + 3*A*b^2*c)*d*e^7)*m^2 - 36*(35*B*c^3*d^4*e^4 - 691*A*b^3*e^8 - 40*(3*B*b*c^2 + A*c^3)*d^3*e
^5 + 140*(B*b^2*c + A*b*c^2)*d^2*e^6 - 56*(B*b^3 + 3*A*b^2*c)*d*e^7)*m)*x^4 + 12*(13*A*b^3*d^4*e^4 - 2*(B*b^3
+ 3*A*b^2*c)*d^5*e^3)*m^3 - (A*b^3*d*e^7*m^7 + (29*A*b^3*d*e^7 - 4*(B*b^3 + 3*A*b^2*c)*d^2*e^6)*m^6 + (331*A*b
^3*d*e^7 + 60*(B*b^2*c + A*b*c^2)*d^3*e^5 - 96*(B*b^3 + 3*A*b^2*c)*d^2*e^6)*m^5 + (1871*A*b^3*d*e^7 - 120*(3*B
*b*c^2 + A*c^3)*d^4*e^4 + 1080*(B*b^2*c + A*b*c^2)*d^3*e^5 - 844*(B*b^3 + 3*A*b^2*c)*d^2*e^6)*m^4 + 4*(210*B*c
^3*d^5*e^3 + 1345*A*b^3*d*e^7 - 330*(3*B*b*c^2 + A*c^3)*d^4*e^4 + 1545*(B*b^2*c + A*b*c^2)*d^3*e^5 - 816*(B*b^
3 + 3*A*b^2*c)*d^2*e^6)*m^3 + 4*(630*B*c^3*d^5*e^3 + 1793*A*b^3*d*e^7 - 780*(3*B*b*c^2 + A*c^3)*d^4*e^4 + 2970
*(B*b^2*c + A*b*c^2)*d^3*e^5 - 1300*(B*b^3 + 3*A*b^2*c)*d^2*e^6)*m^2 + 48*(35*B*c^3*d^5*e^3 + 70*A*b^3*d*e^7 -
40*(3*B*b*c^2 + A*c^3)*d^4*e^4 + 140*(B*b^2*c + A*b*c^2)*d^3*e^5 - 56*(B*b^3 + 3*A*b^2*c)*d^2*e^6)*m)*x^3 + 6
*(251*A*b^3*d^4*e^4 + 60*(B*b^2*c + A*b*c^2)*d^6*e^2 - 84*(B*b^3 + 3*A*b^2*c)*d^5*e^3)*m^2 + 3*(A*b^3*d^2*e^6*
m^6 + (27*A*b^3*d^2*e^6 - 4*(B*b^3 + 3*A*b^2*c)*d^3*e^5)*m^5 + (277*A*b^3*d^2*e^6 + 60*(B*b^2*c + A*b*c^2)*d^4
*e^4 - 88*(B*b^3 + 3*A*b^2*c)*d^3*e^5)*m^4 + (1317*A*b^3*d^2*e^6 - 120*(3*B*b*c^2 + A*c^3)*d^5*e^3 + 960*(B*b^
2*c + A*b*c^2)*d^4*e^4 - 668*(B*b^3 + 3*A*b^2*c)*d^3*e^5)*m^3 + 2*(420*B*c^3*d^6*e^2 + 1373*A*b^3*d^2*e^6 - 54
0*(3*B*b*c^2 + A*c^3)*d^5*e^3 + 2130*(B*b^2*c + A*b*c^2)*d^4*e^4 - 964*(B*b^3 + 3*A*b^2*c)*d^3*e^5)*m^2 + 24*(
35*B*c^3*d^6*e^2 + 70*A*b^3*d^2*e^6 - 40*(3*B*b*c^2 + A*c^3)*d^5*e^3 + 140*(B*b^2*c + A*b*c^2)*d^4*e^4 - 56*(B
*b^3 + 3*A*b^2*c)*d^3*e^5)*m)*x^2 + 12*(533*A*b^3*d^4*e^4 - 60*(3*B*b*c^2 + A*c^3)*d^7*e + 450*(B*b^2*c + A*b*
c^2)*d^6*e^2 - 292*(B*b^3 + 3*A*b^2*c)*d^5*e^3)*m - 6*(A*b^3*d^3*e^5*m^5 + 2*(13*A*b^3*d^3*e^5 - 2*(B*b^3 + 3*
A*b^2*c)*d^4*e^4)*m^4 + (251*A*b^3*d^3*e^5 + 60*(B*b^2*c + A*b*c^2)*d^5*e^3 - 84*(B*b^3 + 3*A*b^2*c)*d^4*e^4)*
m^3 + 2*(533*A*b^3*d^3*e^5 - 60*(3*B*b*c^2 + A*c^3)*d^6*e^2 + 450*(B*b^2*c + A*b*c^2)*d^5*e^3 - 292*(B*b^3 + 3
*A*b^2*c)*d^4*e^4)*m^2 + 24*(35*B*c^3*d^7*e + 70*A*b^3*d^3*e^5 - 40*(3*B*b*c^2 + A*c^3)*d^6*e^2 + 140*(B*b^2*c
+ A*b*c^2)*d^5*e^3 - 56*(B*b^3 + 3*A*b^2*c)*d^4*e^4)*m)*x)*(e*x + d)^m/(e^8*m^8 + 36*e^8*m^7 + 546*e^8*m^6 +
4536*e^8*m^5 + 22449*e^8*m^4 + 67284*e^8*m^3 + 118124*e^8*m^2 + 109584*e^8*m + 40320*e^8)

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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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.40952, size = 8995, normalized size = 18.58 \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)^3,x, algorithm="giac")

[Out]

((x*e + d)^m*B*c^3*m^7*x^8*e^8 + (x*e + d)^m*B*c^3*d*m^7*x^7*e^7 + 3*(x*e + d)^m*B*b*c^2*m^7*x^7*e^8 + (x*e +
d)^m*A*c^3*m^7*x^7*e^8 + 28*(x*e + d)^m*B*c^3*m^6*x^8*e^8 + 3*(x*e + d)^m*B*b*c^2*d*m^7*x^6*e^7 + (x*e + d)^m*
A*c^3*d*m^7*x^6*e^7 + 21*(x*e + d)^m*B*c^3*d*m^6*x^7*e^7 - 7*(x*e + d)^m*B*c^3*d^2*m^6*x^6*e^6 + 3*(x*e + d)^m
*B*b^2*c*m^7*x^6*e^8 + 3*(x*e + d)^m*A*b*c^2*m^7*x^6*e^8 + 87*(x*e + d)^m*B*b*c^2*m^6*x^7*e^8 + 29*(x*e + d)^m
*A*c^3*m^6*x^7*e^8 + 322*(x*e + d)^m*B*c^3*m^5*x^8*e^8 + 3*(x*e + d)^m*B*b^2*c*d*m^7*x^5*e^7 + 3*(x*e + d)^m*A
*b*c^2*d*m^7*x^5*e^7 + 69*(x*e + d)^m*B*b*c^2*d*m^6*x^6*e^7 + 23*(x*e + d)^m*A*c^3*d*m^6*x^6*e^7 + 175*(x*e +
d)^m*B*c^3*d*m^5*x^7*e^7 - 18*(x*e + d)^m*B*b*c^2*d^2*m^6*x^5*e^6 - 6*(x*e + d)^m*A*c^3*d^2*m^6*x^5*e^6 - 105*
(x*e + d)^m*B*c^3*d^2*m^5*x^6*e^6 + 42*(x*e + d)^m*B*c^3*d^3*m^5*x^5*e^5 + (x*e + d)^m*B*b^3*m^7*x^5*e^8 + 3*(
x*e + d)^m*A*b^2*c*m^7*x^5*e^8 + 90*(x*e + d)^m*B*b^2*c*m^6*x^6*e^8 + 90*(x*e + d)^m*A*b*c^2*m^6*x^6*e^8 + 102
9*(x*e + d)^m*B*b*c^2*m^5*x^7*e^8 + 343*(x*e + d)^m*A*c^3*m^5*x^7*e^8 + 1960*(x*e + d)^m*B*c^3*m^4*x^8*e^8 + (
x*e + d)^m*B*b^3*d*m^7*x^4*e^7 + 3*(x*e + d)^m*A*b^2*c*d*m^7*x^4*e^7 + 75*(x*e + d)^m*B*b^2*c*d*m^6*x^5*e^7 +
75*(x*e + d)^m*A*b*c^2*d*m^6*x^5*e^7 + 615*(x*e + d)^m*B*b*c^2*d*m^5*x^6*e^7 + 205*(x*e + d)^m*A*c^3*d*m^5*x^6
*e^7 + 735*(x*e + d)^m*B*c^3*d*m^4*x^7*e^7 - 15*(x*e + d)^m*B*b^2*c*d^2*m^6*x^4*e^6 - 15*(x*e + d)^m*A*b*c^2*d
^2*m^6*x^4*e^6 - 324*(x*e + d)^m*B*b*c^2*d^2*m^5*x^5*e^6 - 108*(x*e + d)^m*A*c^3*d^2*m^5*x^5*e^6 - 595*(x*e +
d)^m*B*c^3*d^2*m^4*x^6*e^6 + 90*(x*e + d)^m*B*b*c^2*d^3*m^5*x^4*e^5 + 30*(x*e + d)^m*A*c^3*d^3*m^5*x^4*e^5 + 4
20*(x*e + d)^m*B*c^3*d^3*m^4*x^5*e^5 - 210*(x*e + d)^m*B*c^3*d^4*m^4*x^4*e^4 + (x*e + d)^m*A*b^3*m^7*x^4*e^8 +
31*(x*e + d)^m*B*b^3*m^6*x^5*e^8 + 93*(x*e + d)^m*A*b^2*c*m^6*x^5*e^8 + 1098*(x*e + d)^m*B*b^2*c*m^5*x^6*e^8
+ 1098*(x*e + d)^m*A*b*c^2*m^5*x^6*e^8 + 6405*(x*e + d)^m*B*b*c^2*m^4*x^7*e^8 + 2135*(x*e + d)^m*A*c^3*m^4*x^7
*e^8 + 6769*(x*e + d)^m*B*c^3*m^3*x^8*e^8 + (x*e + d)^m*A*b^3*d*m^7*x^3*e^7 + 27*(x*e + d)^m*B*b^3*d*m^6*x^4*e
^7 + 81*(x*e + d)^m*A*b^2*c*d*m^6*x^4*e^7 + 723*(x*e + d)^m*B*b^2*c*d*m^5*x^5*e^7 + 723*(x*e + d)^m*A*b*c^2*d*
m^5*x^5*e^7 + 2715*(x*e + d)^m*B*b*c^2*d*m^4*x^6*e^7 + 905*(x*e + d)^m*A*c^3*d*m^4*x^6*e^7 + 1624*(x*e + d)^m*
B*c^3*d*m^3*x^7*e^7 - 4*(x*e + d)^m*B*b^3*d^2*m^6*x^3*e^6 - 12*(x*e + d)^m*A*b^2*c*d^2*m^6*x^3*e^6 - 315*(x*e
+ d)^m*B*b^2*c*d^2*m^5*x^4*e^6 - 315*(x*e + d)^m*A*b*c^2*d^2*m^5*x^4*e^6 - 2070*(x*e + d)^m*B*b*c^2*d^2*m^4*x^
5*e^6 - 690*(x*e + d)^m*A*c^3*d^2*m^4*x^5*e^6 - 1575*(x*e + d)^m*B*c^3*d^2*m^3*x^6*e^6 + 60*(x*e + d)^m*B*b^2*
c*d^3*m^5*x^3*e^5 + 60*(x*e + d)^m*A*b*c^2*d^3*m^5*x^3*e^5 + 1260*(x*e + d)^m*B*b*c^2*d^3*m^4*x^4*e^5 + 420*(x
*e + d)^m*A*c^3*d^3*m^4*x^4*e^5 + 1470*(x*e + d)^m*B*c^3*d^3*m^3*x^5*e^5 - 360*(x*e + d)^m*B*b*c^2*d^4*m^4*x^3
*e^4 - 120*(x*e + d)^m*A*c^3*d^4*m^4*x^3*e^4 - 1260*(x*e + d)^m*B*c^3*d^4*m^3*x^4*e^4 + 840*(x*e + d)^m*B*c^3*
d^5*m^3*x^3*e^3 + 32*(x*e + d)^m*A*b^3*m^6*x^4*e^8 + 391*(x*e + d)^m*B*b^3*m^5*x^5*e^8 + 1173*(x*e + d)^m*A*b^
2*c*m^5*x^5*e^8 + 7020*(x*e + d)^m*B*b^2*c*m^4*x^6*e^8 + 7020*(x*e + d)^m*A*b*c^2*m^4*x^6*e^8 + 22512*(x*e + d
)^m*B*b*c^2*m^3*x^7*e^8 + 7504*(x*e + d)^m*A*c^3*m^3*x^7*e^8 + 13132*(x*e + d)^m*B*c^3*m^2*x^8*e^8 + 29*(x*e +
d)^m*A*b^3*d*m^6*x^3*e^7 + 283*(x*e + d)^m*B*b^3*d*m^5*x^4*e^7 + 849*(x*e + d)^m*A*b^2*c*d*m^5*x^4*e^7 + 3405
*(x*e + d)^m*B*b^2*c*d*m^4*x^5*e^7 + 3405*(x*e + d)^m*A*b*c^2*d*m^4*x^5*e^7 + 6222*(x*e + d)^m*B*b*c^2*d*m^3*x
^6*e^7 + 2074*(x*e + d)^m*A*c^3*d*m^3*x^6*e^7 + 1764*(x*e + d)^m*B*c^3*d*m^2*x^7*e^7 - 3*(x*e + d)^m*A*b^3*d^2
*m^6*x^2*e^6 - 96*(x*e + d)^m*B*b^3*d^2*m^5*x^3*e^6 - 288*(x*e + d)^m*A*b^2*c*d^2*m^5*x^3*e^6 - 2355*(x*e + d)
^m*B*b^2*c*d^2*m^4*x^4*e^6 - 2355*(x*e + d)^m*A*b*c^2*d^2*m^4*x^4*e^6 - 5940*(x*e + d)^m*B*b*c^2*d^2*m^3*x^5*e
^6 - 1980*(x*e + d)^m*A*c^3*d^2*m^3*x^5*e^6 - 1918*(x*e + d)^m*B*c^3*d^2*m^2*x^6*e^6 + 12*(x*e + d)^m*B*b^3*d^
3*m^5*x^2*e^5 + 36*(x*e + d)^m*A*b^2*c*d^3*m^5*x^2*e^5 + 1080*(x*e + d)^m*B*b^2*c*d^3*m^4*x^3*e^5 + 1080*(x*e
+ d)^m*A*b*c^2*d^3*m^4*x^3*e^5 + 5310*(x*e + d)^m*B*b*c^2*d^3*m^3*x^4*e^5 + 1770*(x*e + d)^m*A*c^3*d^3*m^3*x^4
*e^5 + 2100*(x*e + d)^m*B*c^3*d^3*m^2*x^5*e^5 - 180*(x*e + d)^m*B*b^2*c*d^4*m^4*x^2*e^4 - 180*(x*e + d)^m*A*b*
c^2*d^4*m^4*x^2*e^4 - 3960*(x*e + d)^m*B*b*c^2*d^4*m^3*x^3*e^4 - 1320*(x*e + d)^m*A*c^3*d^4*m^3*x^3*e^4 - 2310
*(x*e + d)^m*B*c^3*d^4*m^2*x^4*e^4 + 1080*(x*e + d)^m*B*b*c^2*d^5*m^3*x^2*e^3 + 360*(x*e + d)^m*A*c^3*d^5*m^3*
x^2*e^3 + 2520*(x*e + d)^m*B*c^3*d^5*m^2*x^3*e^3 - 2520*(x*e + d)^m*B*c^3*d^6*m^2*x^2*e^2 + 418*(x*e + d)^m*A*
b^3*m^5*x^4*e^8 + 2581*(x*e + d)^m*B*b^3*m^4*x^5*e^8 + 7743*(x*e + d)^m*A*b^2*c*m^4*x^5*e^8 + 25227*(x*e + d)^
m*B*b^2*c*m^3*x^6*e^8 + 25227*(x*e + d)^m*A*b*c^2*m^3*x^6*e^8 + 44268*(x*e + d)^m*B*b*c^2*m^2*x^7*e^8 + 14756*
(x*e + d)^m*A*c^3*m^2*x^7*e^8 + 13068*(x*e + d)^m*B*c^3*m*x^8*e^8 + 331*(x*e + d)^m*A*b^3*d*m^5*x^3*e^7 + 1449
*(x*e + d)^m*B*b^3*d*m^4*x^4*e^7 + 4347*(x*e + d)^m*A*b^2*c*d*m^4*x^4*e^7 + 8202*(x*e + d)^m*B*b^2*c*d*m^3*x^5
*e^7 + 8202*(x*e + d)^m*A*b*c^2*d*m^3*x^5*e^7 + 6936*(x*e + d)^m*B*b*c^2*d*m^2*x^6*e^7 + 2312*(x*e + d)^m*A*c^
3*d*m^2*x^6*e^7 + 720*(x*e + d)^m*B*c^3*d*m*x^7*e^7 - 81*(x*e + d)^m*A*b^3*d^2*m^5*x^2*e^6 - 844*(x*e + d)^m*B
*b^3*d^2*m^4*x^3*e^6 - 2532*(x*e + d)^m*A*b^2*c*d^2*m^4*x^3*e^6 - 7605*(x*e + d)^m*B*b^2*c*d^2*m^3*x^4*e^6 - 7
605*(x*e + d)^m*A*b*c^2*d^2*m^3*x^4*e^6 - 7632*(x*e + d)^m*B*b*c^2*d^2*m^2*x^5*e^6 - 2544*(x*e + d)^m*A*c^3*d^
2*m^2*x^5*e^6 - 840*(x*e + d)^m*B*c^3*d^2*m*x^6*e^6 + 6*(x*e + d)^m*A*b^3*d^3*m^5*x*e^5 + 264*(x*e + d)^m*B*b^
3*d^3*m^4*x^2*e^5 + 792*(x*e + d)^m*A*b^2*c*d^3*m^4*x^2*e^5 + 6180*(x*e + d)^m*B*b^2*c*d^3*m^3*x^3*e^5 + 6180*
(x*e + d)^m*A*b*c^2*d^3*m^3*x^3*e^5 + 8460*(x*e + d)^m*B*b*c^2*d^3*m^2*x^4*e^5 + 2820*(x*e + d)^m*A*c^3*d^3*m^
2*x^4*e^5 + 1008*(x*e + d)^m*B*c^3*d^3*m*x^5*e^5 - 24*(x*e + d)^m*B*b^3*d^4*m^4*x*e^4 - 72*(x*e + d)^m*A*b^2*c
*d^4*m^4*x*e^4 - 2880*(x*e + d)^m*B*b^2*c*d^4*m^3*x^2*e^4 - 2880*(x*e + d)^m*A*b*c^2*d^4*m^3*x^2*e^4 - 9360*(x
*e + d)^m*B*b*c^2*d^4*m^2*x^3*e^4 - 3120*(x*e + d)^m*A*c^3*d^4*m^2*x^3*e^4 - 1260*(x*e + d)^m*B*c^3*d^4*m*x^4*
e^4 + 360*(x*e + d)^m*B*b^2*c*d^5*m^3*x*e^3 + 360*(x*e + d)^m*A*b*c^2*d^5*m^3*x*e^3 + 9720*(x*e + d)^m*B*b*c^2
*d^5*m^2*x^2*e^3 + 3240*(x*e + d)^m*A*c^3*d^5*m^2*x^2*e^3 + 1680*(x*e + d)^m*B*c^3*d^5*m*x^3*e^3 - 2160*(x*e +
d)^m*B*b*c^2*d^6*m^2*x*e^2 - 720*(x*e + d)^m*A*c^3*d^6*m^2*x*e^2 - 2520*(x*e + d)^m*B*c^3*d^6*m*x^2*e^2 + 504
0*(x*e + d)^m*B*c^3*d^7*m*x*e + 2864*(x*e + d)^m*A*b^3*m^4*x^4*e^8 + 9544*(x*e + d)^m*B*b^3*m^3*x^5*e^8 + 2863
2*(x*e + d)^m*A*b^2*c*m^3*x^5*e^8 + 50490*(x*e + d)^m*B*b^2*c*m^2*x^6*e^8 + 50490*(x*e + d)^m*A*b*c^2*m^2*x^6*
e^8 + 44496*(x*e + d)^m*B*b*c^2*m*x^7*e^8 + 14832*(x*e + d)^m*A*c^3*m*x^7*e^8 + 5040*(x*e + d)^m*B*c^3*x^8*e^8
+ 1871*(x*e + d)^m*A*b^3*d*m^4*x^3*e^7 + 3748*(x*e + d)^m*B*b^3*d*m^3*x^4*e^7 + 11244*(x*e + d)^m*A*b^2*c*d*m
^3*x^4*e^7 + 9480*(x*e + d)^m*B*b^2*c*d*m^2*x^5*e^7 + 9480*(x*e + d)^m*A*b*c^2*d*m^2*x^5*e^7 + 2880*(x*e + d)^
m*B*b*c^2*d*m*x^6*e^7 + 960*(x*e + d)^m*A*c^3*d*m*x^6*e^7 - 831*(x*e + d)^m*A*b^3*d^2*m^4*x^2*e^6 - 3264*(x*e
+ d)^m*B*b^3*d^2*m^3*x^3*e^6 - 9792*(x*e + d)^m*A*b^2*c*d^2*m^3*x^3*e^6 - 10590*(x*e + d)^m*B*b^2*c*d^2*m^2*x^
4*e^6 - 10590*(x*e + d)^m*A*b*c^2*d^2*m^2*x^4*e^6 - 3456*(x*e + d)^m*B*b*c^2*d^2*m*x^5*e^6 - 1152*(x*e + d)^m*
A*c^3*d^2*m*x^5*e^6 + 156*(x*e + d)^m*A*b^3*d^3*m^4*x*e^5 + 2004*(x*e + d)^m*B*b^3*d^3*m^3*x^2*e^5 + 6012*(x*e
+ d)^m*A*b^2*c*d^3*m^3*x^2*e^5 + 11880*(x*e + d)^m*B*b^2*c*d^3*m^2*x^3*e^5 + 11880*(x*e + d)^m*A*b*c^2*d^3*m^
2*x^3*e^5 + 4320*(x*e + d)^m*B*b*c^2*d^3*m*x^4*e^5 + 1440*(x*e + d)^m*A*c^3*d^3*m*x^4*e^5 - 6*(x*e + d)^m*A*b^
3*d^4*m^4*e^4 - 504*(x*e + d)^m*B*b^3*d^4*m^3*x*e^4 - 1512*(x*e + d)^m*A*b^2*c*d^4*m^3*x*e^4 - 12780*(x*e + d)
^m*B*b^2*c*d^4*m^2*x^2*e^4 - 12780*(x*e + d)^m*A*b*c^2*d^4*m^2*x^2*e^4 - 5760*(x*e + d)^m*B*b*c^2*d^4*m*x^3*e^
4 - 1920*(x*e + d)^m*A*c^3*d^4*m*x^3*e^4 + 24*(x*e + d)^m*B*b^3*d^5*m^3*e^3 + 72*(x*e + d)^m*A*b^2*c*d^5*m^3*e
^3 + 5400*(x*e + d)^m*B*b^2*c*d^5*m^2*x*e^3 + 5400*(x*e + d)^m*A*b*c^2*d^5*m^2*x*e^3 + 8640*(x*e + d)^m*B*b*c^
2*d^5*m*x^2*e^3 + 2880*(x*e + d)^m*A*c^3*d^5*m*x^2*e^3 - 360*(x*e + d)^m*B*b^2*c*d^6*m^2*e^2 - 360*(x*e + d)^m
*A*b*c^2*d^6*m^2*e^2 - 17280*(x*e + d)^m*B*b*c^2*d^6*m*x*e^2 - 5760*(x*e + d)^m*A*c^3*d^6*m*x*e^2 + 2160*(x*e
+ d)^m*B*b*c^2*d^7*m*e + 720*(x*e + d)^m*A*c^3*d^7*m*e - 5040*(x*e + d)^m*B*c^3*d^8 + 10993*(x*e + d)^m*A*b^3*
m^3*x^4*e^8 + 19564*(x*e + d)^m*B*b^3*m^2*x^5*e^8 + 58692*(x*e + d)^m*A*b^2*c*m^2*x^5*e^8 + 51432*(x*e + d)^m*
B*b^2*c*m*x^6*e^8 + 51432*(x*e + d)^m*A*b*c^2*m*x^6*e^8 + 17280*(x*e + d)^m*B*b*c^2*x^7*e^8 + 5760*(x*e + d)^m
*A*c^3*x^7*e^8 + 5380*(x*e + d)^m*A*b^3*d*m^3*x^3*e^7 + 4572*(x*e + d)^m*B*b^3*d*m^2*x^4*e^7 + 13716*(x*e + d)
^m*A*b^2*c*d*m^2*x^4*e^7 + 4032*(x*e + d)^m*B*b^2*c*d*m*x^5*e^7 + 4032*(x*e + d)^m*A*b*c^2*d*m*x^5*e^7 - 3951*
(x*e + d)^m*A*b^3*d^2*m^3*x^2*e^6 - 5200*(x*e + d)^m*B*b^3*d^2*m^2*x^3*e^6 - 15600*(x*e + d)^m*A*b^2*c*d^2*m^2
*x^3*e^6 - 5040*(x*e + d)^m*B*b^2*c*d^2*m*x^4*e^6 - 5040*(x*e + d)^m*A*b*c^2*d^2*m*x^4*e^6 + 1506*(x*e + d)^m*
A*b^3*d^3*m^3*x*e^5 + 5784*(x*e + d)^m*B*b^3*d^3*m^2*x^2*e^5 + 17352*(x*e + d)^m*A*b^2*c*d^3*m^2*x^2*e^5 + 672
0*(x*e + d)^m*B*b^2*c*d^3*m*x^3*e^5 + 6720*(x*e + d)^m*A*b*c^2*d^3*m*x^3*e^5 - 156*(x*e + d)^m*A*b^3*d^4*m^3*e
^4 - 3504*(x*e + d)^m*B*b^3*d^4*m^2*x*e^4 - 10512*(x*e + d)^m*A*b^2*c*d^4*m^2*x*e^4 - 10080*(x*e + d)^m*B*b^2*
c*d^4*m*x^2*e^4 - 10080*(x*e + d)^m*A*b*c^2*d^4*m*x^2*e^4 + 504*(x*e + d)^m*B*b^3*d^5*m^2*e^3 + 1512*(x*e + d)
^m*A*b^2*c*d^5*m^2*e^3 + 20160*(x*e + d)^m*B*b^2*c*d^5*m*x*e^3 + 20160*(x*e + d)^m*A*b*c^2*d^5*m*x*e^3 - 5400*
(x*e + d)^m*B*b^2*c*d^6*m*e^2 - 5400*(x*e + d)^m*A*b*c^2*d^6*m*e^2 + 17280*(x*e + d)^m*B*b*c^2*d^7*e + 5760*(x
*e + d)^m*A*c^3*d^7*e + 23312*(x*e + d)^m*A*b^3*m^2*x^4*e^8 + 20304*(x*e + d)^m*B*b^3*m*x^5*e^8 + 60912*(x*e +
d)^m*A*b^2*c*m*x^5*e^8 + 20160*(x*e + d)^m*B*b^2*c*x^6*e^8 + 20160*(x*e + d)^m*A*b*c^2*x^6*e^8 + 7172*(x*e +
d)^m*A*b^3*d*m^2*x^3*e^7 + 2016*(x*e + d)^m*B*b^3*d*m*x^4*e^7 + 6048*(x*e + d)^m*A*b^2*c*d*m*x^4*e^7 - 8238*(x
*e + d)^m*A*b^3*d^2*m^2*x^2*e^6 - 2688*(x*e + d)^m*B*b^3*d^2*m*x^3*e^6 - 8064*(x*e + d)^m*A*b^2*c*d^2*m*x^3*e^
6 + 6396*(x*e + d)^m*A*b^3*d^3*m^2*x*e^5 + 4032*(x*e + d)^m*B*b^3*d^3*m*x^2*e^5 + 12096*(x*e + d)^m*A*b^2*c*d^
3*m*x^2*e^5 - 1506*(x*e + d)^m*A*b^3*d^4*m^2*e^4 - 8064*(x*e + d)^m*B*b^3*d^4*m*x*e^4 - 24192*(x*e + d)^m*A*b^
2*c*d^4*m*x*e^4 + 3504*(x*e + d)^m*B*b^3*d^5*m*e^3 + 10512*(x*e + d)^m*A*b^2*c*d^5*m*e^3 - 20160*(x*e + d)^m*B
*b^2*c*d^6*e^2 - 20160*(x*e + d)^m*A*b*c^2*d^6*e^2 + 24876*(x*e + d)^m*A*b^3*m*x^4*e^8 + 8064*(x*e + d)^m*B*b^
3*x^5*e^8 + 24192*(x*e + d)^m*A*b^2*c*x^5*e^8 + 3360*(x*e + d)^m*A*b^3*d*m*x^3*e^7 - 5040*(x*e + d)^m*A*b^3*d^
2*m*x^2*e^6 + 10080*(x*e + d)^m*A*b^3*d^3*m*x*e^5 - 6396*(x*e + d)^m*A*b^3*d^4*m*e^4 + 8064*(x*e + d)^m*B*b^3*
d^5*e^3 + 24192*(x*e + d)^m*A*b^2*c*d^5*e^3 + 10080*(x*e + d)^m*A*b^3*x^4*e^8 - 10080*(x*e + d)^m*A*b^3*d^4*e^
4)/(m^8*e^8 + 36*m^7*e^8 + 546*m^6*e^8 + 4536*m^5*e^8 + 22449*m^4*e^8 + 67284*m^3*e^8 + 118124*m^2*e^8 + 10958
4*m*e^8 + 40320*e^8)