3.2146 \(\int (a+b x) (d+e x)^m (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=239 \[ -\frac{21 b^2 (b d-a e)^5 (d+e x)^{m+3}}{e^8 (m+3)}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{m+4}}{e^8 (m+4)}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{m+5}}{e^8 (m+5)}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{m+6}}{e^8 (m+6)}-\frac{7 b^6 (b d-a e) (d+e x)^{m+7}}{e^8 (m+7)}-\frac{(b d-a e)^7 (d+e x)^{m+1}}{e^8 (m+1)}+\frac{7 b (b d-a e)^6 (d+e x)^{m+2}}{e^8 (m+2)}+\frac{b^7 (d+e x)^{m+8}}{e^8 (m+8)} \]

[Out]

-(((b*d - a*e)^7*(d + e*x)^(1 + m))/(e^8*(1 + m))) + (7*b*(b*d - a*e)^6*(d + e*x)^(2 + m))/(e^8*(2 + m)) - (21
*b^2*(b*d - a*e)^5*(d + e*x)^(3 + m))/(e^8*(3 + m)) + (35*b^3*(b*d - a*e)^4*(d + e*x)^(4 + m))/(e^8*(4 + m)) -
(35*b^4*(b*d - a*e)^3*(d + e*x)^(5 + m))/(e^8*(5 + m)) + (21*b^5*(b*d - a*e)^2*(d + e*x)^(6 + m))/(e^8*(6 + m
)) - (7*b^6*(b*d - a*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (b^7*(d + e*x)^(8 + m))/(e^8*(8 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.139468, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac{21 b^2 (b d-a e)^5 (d+e x)^{m+3}}{e^8 (m+3)}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{m+4}}{e^8 (m+4)}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{m+5}}{e^8 (m+5)}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{m+6}}{e^8 (m+6)}-\frac{7 b^6 (b d-a e) (d+e x)^{m+7}}{e^8 (m+7)}-\frac{(b d-a e)^7 (d+e x)^{m+1}}{e^8 (m+1)}+\frac{7 b (b d-a e)^6 (d+e x)^{m+2}}{e^8 (m+2)}+\frac{b^7 (d+e x)^{m+8}}{e^8 (m+8)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b*d - a*e)^7*(d + e*x)^(1 + m))/(e^8*(1 + m))) + (7*b*(b*d - a*e)^6*(d + e*x)^(2 + m))/(e^8*(2 + m)) - (21
*b^2*(b*d - a*e)^5*(d + e*x)^(3 + m))/(e^8*(3 + m)) + (35*b^3*(b*d - a*e)^4*(d + e*x)^(4 + m))/(e^8*(4 + m)) -
(35*b^4*(b*d - a*e)^3*(d + e*x)^(5 + m))/(e^8*(5 + m)) + (21*b^5*(b*d - a*e)^2*(d + e*x)^(6 + m))/(e^8*(6 + m
)) - (7*b^6*(b*d - a*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (b^7*(d + e*x)^(8 + m))/(e^8*(8 + m))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^m \, dx\\ &=\int \left (\frac{(-b d+a e)^7 (d+e x)^m}{e^7}+\frac{7 b (b d-a e)^6 (d+e x)^{1+m}}{e^7}-\frac{21 b^2 (b d-a e)^5 (d+e x)^{2+m}}{e^7}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{3+m}}{e^7}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{4+m}}{e^7}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{5+m}}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^{6+m}}{e^7}+\frac{b^7 (d+e x)^{7+m}}{e^7}\right ) \, dx\\ &=-\frac{(b d-a e)^7 (d+e x)^{1+m}}{e^8 (1+m)}+\frac{7 b (b d-a e)^6 (d+e x)^{2+m}}{e^8 (2+m)}-\frac{21 b^2 (b d-a e)^5 (d+e x)^{3+m}}{e^8 (3+m)}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{4+m}}{e^8 (4+m)}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{5+m}}{e^8 (5+m)}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{6+m}}{e^8 (6+m)}-\frac{7 b^6 (b d-a e) (d+e x)^{7+m}}{e^8 (7+m)}+\frac{b^7 (d+e x)^{8+m}}{e^8 (8+m)}\\ \end{align*}

Mathematica [A]  time = 0.22813, size = 203, normalized size = 0.85 \[ \frac{(d+e x)^{m+1} \left (-\frac{21 b^2 (d+e x)^2 (b d-a e)^5}{m+3}+\frac{35 b^3 (d+e x)^3 (b d-a e)^4}{m+4}-\frac{35 b^4 (d+e x)^4 (b d-a e)^3}{m+5}+\frac{21 b^5 (d+e x)^5 (b d-a e)^2}{m+6}-\frac{7 b^6 (d+e x)^6 (b d-a e)}{m+7}+\frac{7 b (d+e x) (b d-a e)^6}{m+2}-\frac{(b d-a e)^7}{m+1}+\frac{b^7 (d+e x)^7}{m+8}\right )}{e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(-((b*d - a*e)^7/(1 + m)) + (7*b*(b*d - a*e)^6*(d + e*x))/(2 + m) - (21*b^2*(b*d - a*e)^5*(
d + e*x)^2)/(3 + m) + (35*b^3*(b*d - a*e)^4*(d + e*x)^3)/(4 + m) - (35*b^4*(b*d - a*e)^3*(d + e*x)^4)/(5 + m)
+ (21*b^5*(b*d - a*e)^2*(d + e*x)^5)/(6 + m) - (7*b^6*(b*d - a*e)*(d + e*x)^6)/(7 + m) + (b^7*(d + e*x)^7)/(8
+ m)))/e^8

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 3244, normalized size = 13.6 \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*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

(e*x+d)^(1+m)*(b^7*e^7*m^7*x^7+7*a*b^6*e^7*m^7*x^6+28*b^7*e^7*m^6*x^7+21*a^2*b^5*e^7*m^7*x^5+203*a*b^6*e^7*m^6
*x^6-7*b^7*d*e^6*m^6*x^6+322*b^7*e^7*m^5*x^7+35*a^3*b^4*e^7*m^7*x^4+630*a^2*b^5*e^7*m^6*x^5-42*a*b^6*d*e^6*m^6
*x^5+2401*a*b^6*e^7*m^5*x^6-147*b^7*d*e^6*m^5*x^6+1960*b^7*e^7*m^4*x^7+35*a^4*b^3*e^7*m^7*x^3+1085*a^3*b^4*e^7
*m^6*x^4-105*a^2*b^5*d*e^6*m^6*x^4+7686*a^2*b^5*e^7*m^5*x^5-966*a*b^6*d*e^6*m^5*x^5+14945*a*b^6*e^7*m^4*x^6+42
*b^7*d^2*e^5*m^5*x^5-1225*b^7*d*e^6*m^4*x^6+6769*b^7*e^7*m^3*x^7+21*a^5*b^2*e^7*m^7*x^2+1120*a^4*b^3*e^7*m^6*x
^3-140*a^3*b^4*d*e^6*m^6*x^3+13685*a^3*b^4*e^7*m^5*x^4-2625*a^2*b^5*d*e^6*m^5*x^4+49140*a^2*b^5*e^7*m^4*x^5+21
0*a*b^6*d^2*e^5*m^5*x^4-8610*a*b^6*d*e^6*m^4*x^5+52528*a*b^6*e^7*m^3*x^6+630*b^7*d^2*e^5*m^4*x^5-5145*b^7*d*e^
6*m^3*x^6+13132*b^7*e^7*m^2*x^7+7*a^6*b*e^7*m^7*x+693*a^5*b^2*e^7*m^6*x^2-105*a^4*b^3*d*e^6*m^6*x^2+14630*a^4*
b^3*e^7*m^5*x^3-3780*a^3*b^4*d*e^6*m^5*x^3+90335*a^3*b^4*e^7*m^4*x^4+420*a^2*b^5*d^2*e^5*m^5*x^3-25305*a^2*b^5
*d*e^6*m^4*x^4+176589*a^2*b^5*e^7*m^3*x^5+3780*a*b^6*d^2*e^5*m^4*x^4-38010*a*b^6*d*e^6*m^3*x^5+103292*a*b^6*e^
7*m^2*x^6-210*b^7*d^3*e^4*m^4*x^4+3570*b^7*d^2*e^5*m^3*x^5-11368*b^7*d*e^6*m^2*x^6+13068*b^7*e^7*m*x^7+a^7*e^7
*m^7+238*a^6*b*e^7*m^6*x-42*a^5*b^2*d*e^6*m^6*x+9387*a^5*b^2*e^7*m^5*x^2-3045*a^4*b^3*d*e^6*m^5*x^2+100240*a^4
*b^3*e^7*m^4*x^3+420*a^3*b^4*d^2*e^5*m^5*x^2-39620*a^3*b^4*d*e^6*m^4*x^3+334040*a^3*b^4*e^7*m^3*x^4+8820*a^2*b
^5*d^2*e^5*m^4*x^3-119175*a^2*b^5*d*e^6*m^3*x^4+353430*a^2*b^5*e^7*m^2*x^5-840*a*b^6*d^3*e^4*m^4*x^3+24150*a*b
^6*d^2*e^5*m^3*x^4-87108*a*b^6*d*e^6*m^2*x^5+103824*a*b^6*e^7*m*x^6-2100*b^7*d^3*e^4*m^3*x^4+9450*b^7*d^2*e^5*
m^2*x^5-12348*b^7*d*e^6*m*x^6+5040*b^7*e^7*x^7+35*a^7*e^7*m^6-7*a^6*b*d*e^6*m^6+3346*a^6*b*e^7*m^5*x-1302*a^5*
b^2*d*e^6*m^5*x+67095*a^5*b^2*e^7*m^4*x^2+210*a^4*b^3*d^2*e^5*m^5*x-34755*a^4*b^3*d*e^6*m^4*x^2+384755*a^4*b^3
*e^7*m^3*x^3+10080*a^3*b^4*d^2*e^5*m^4*x^2-202860*a^3*b^4*d*e^6*m^3*x^3+684740*a^3*b^4*e^7*m^2*x^4-1260*a^2*b^
5*d^3*e^4*m^4*x^2+65940*a^2*b^5*d^2*e^5*m^3*x^3-287070*a^2*b^5*d*e^6*m^2*x^4+360024*a^2*b^5*e^7*m*x^5-11760*a*
b^6*d^3*e^4*m^3*x^3+69300*a*b^6*d^2*e^5*m^2*x^4-97104*a*b^6*d*e^6*m*x^5+40320*a*b^6*e^7*x^6+840*b^7*d^4*e^3*m^
3*x^3-7350*b^7*d^3*e^4*m^2*x^4+11508*b^7*d^2*e^5*m*x^5-5040*b^7*d*e^6*x^6+511*a^7*e^7*m^5-231*a^6*b*d*e^6*m^5+
25060*a^6*b*e^7*m^4*x+42*a^5*b^2*d^2*e^5*m^5-16170*a^5*b^2*d*e^6*m^4*x+270144*a^5*b^2*e^7*m^3*x^2+5670*a^4*b^3
*d^2*e^5*m^4*x-196455*a^4*b^3*d*e^6*m^3*x^2+815920*a^4*b^3*e^7*m^2*x^3-840*a^3*b^4*d^3*e^4*m^4*x+88620*a^3*b^4
*d^2*e^5*m^3*x^2-524720*a^3*b^4*d*e^6*m^2*x^3+710640*a^3*b^4*e^7*m*x^4-22680*a^2*b^5*d^3*e^4*m^3*x^2+212940*a^
2*b^5*d^2*e^5*m^2*x^3-331800*a^2*b^5*d*e^6*m*x^4+141120*a^2*b^5*e^7*x^5+2520*a*b^6*d^4*e^3*m^3*x^2-49560*a*b^6
*d^3*e^4*m^2*x^3+89040*a*b^6*d^2*e^5*m*x^4-40320*a*b^6*d*e^6*x^5+5040*b^7*d^4*e^3*m^2*x^3-10500*b^7*d^3*e^4*m*
x^4+5040*b^7*d^2*e^5*x^5+4025*a^7*e^7*m^4-3115*a^6*b*d*e^6*m^4+107023*a^6*b*e^7*m^3*x+1260*a^5*b^2*d^2*e^5*m^4
-101850*a^5*b^2*d*e^6*m^3*x+602532*a^5*b^2*e^7*m^2*x^2-210*a^4*b^3*d^3*e^4*m^4+58170*a^4*b^3*d^2*e^5*m^3*x-564
900*a^4*b^3*d*e^6*m^2*x^2+870660*a^4*b^3*e^7*m*x^3-18480*a^3*b^4*d^3*e^4*m^3*x+342720*a^3*b^4*d^2*e^5*m^2*x^2-
640080*a^3*b^4*d*e^6*m*x^3+282240*a^3*b^4*e^7*x^4+2520*a^2*b^5*d^4*e^3*m^3*x-129780*a^2*b^5*d^3*e^4*m^2*x^2+29
6520*a^2*b^5*d^2*e^5*m*x^3-141120*a^2*b^5*d*e^6*x^4+27720*a*b^6*d^4*e^3*m^2*x^2-78960*a*b^6*d^3*e^4*m*x^3+4032
0*a*b^6*d^2*e^5*x^4-2520*b^7*d^5*e^2*m^2*x^2+9240*b^7*d^4*e^3*m*x^3-5040*b^7*d^3*e^4*x^4+18424*a^7*e^7*m^3-219
45*a^6*b*d*e^6*m^3+256942*a^6*b*e^7*m^2*x+14910*a^5*b^2*d^2*e^5*m^3-336588*a^5*b^2*d*e^6*m^2*x+673008*a^5*b^2*
e^7*m*x^2-5460*a^4*b^3*d^3*e^4*m^3+276570*a^4*b^3*d^2*e^5*m^2*x-753060*a^4*b^3*d*e^6*m*x^2+352800*a^4*b^3*e^7*
x^3+840*a^3*b^4*d^4*e^3*m^3-140280*a^3*b^4*d^3*e^4*m^2*x+546000*a^3*b^4*d^2*e^5*m*x^2-282240*a^3*b^4*d*e^6*x^3
+40320*a^2*b^5*d^4*e^3*m^2*x-249480*a^2*b^5*d^3*e^4*m*x^2+141120*a^2*b^5*d^2*e^5*x^3-5040*a*b^6*d^5*e^2*m^2*x+
65520*a*b^6*d^4*e^3*m*x^2-40320*a*b^6*d^3*e^4*x^3-7560*b^7*d^5*e^2*m*x^2+5040*b^7*d^4*e^3*x^3+48860*a^7*e^7*m^
2-85078*a^6*b*d*e^6*m^2+312984*a^6*b*e^7*m*x+86940*a^5*b^2*d^2*e^5*m^2-531888*a^5*b^2*d*e^6*m*x+282240*a^5*b^2
*e^7*x^2-52710*a^4*b^3*d^3*e^4*m^2+576660*a^4*b^3*d^2*e^5*m*x-352800*a^4*b^3*d*e^6*x^2+17640*a^3*b^4*d^4*e^3*m
^2-404880*a^3*b^4*d^3*e^4*m*x+282240*a^3*b^4*d^2*e^5*x^2-2520*a^2*b^5*d^5*e^2*m^2+178920*a^2*b^5*d^4*e^3*m*x-1
41120*a^2*b^5*d^3*e^4*x^2-45360*a*b^6*d^5*e^2*m*x+40320*a*b^6*d^4*e^3*x^2+5040*b^7*d^6*e*m*x-5040*b^7*d^5*e^2*
x^2+69264*a^7*e^7*m-171864*a^6*b*d*e^6*m+141120*a^6*b*e^7*x+249648*a^5*b^2*d^2*e^5*m-282240*a^5*b^2*d*e^6*x-22
3860*a^4*b^3*d^3*e^4*m+352800*a^4*b^3*d^2*e^5*x+122640*a^3*b^4*d^4*e^3*m-282240*a^3*b^4*d^3*e^4*x-37800*a^2*b^
5*d^5*e^2*m+141120*a^2*b^5*d^4*e^3*x+5040*a*b^6*d^6*e*m-40320*a*b^6*d^5*e^2*x+5040*b^7*d^6*e*x+40320*a^7*e^7-1
41120*a^6*b*d*e^6+282240*a^5*b^2*d^2*e^5-352800*a^4*b^3*d^3*e^4+282240*a^3*b^4*d^4*e^3-141120*a^2*b^5*d^5*e^2+
40320*a*b^6*d^6*e-5040*b^7*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 [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)^m*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.34125, size = 7007, normalized size = 29.32 \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*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

(a^7*d*e^7*m^7 - 5040*b^7*d^8 + 40320*a*b^6*d^7*e - 141120*a^2*b^5*d^6*e^2 + 282240*a^3*b^4*d^5*e^3 - 352800*a
^4*b^3*d^4*e^4 + 282240*a^5*b^2*d^3*e^5 - 141120*a^6*b*d^2*e^6 + 40320*a^7*d*e^7 + (b^7*e^8*m^7 + 28*b^7*e^8*m
^6 + 322*b^7*e^8*m^5 + 1960*b^7*e^8*m^4 + 6769*b^7*e^8*m^3 + 13132*b^7*e^8*m^2 + 13068*b^7*e^8*m + 5040*b^7*e^
8)*x^8 + (40320*a*b^6*e^8 + (b^7*d*e^7 + 7*a*b^6*e^8)*m^7 + 7*(3*b^7*d*e^7 + 29*a*b^6*e^8)*m^6 + 7*(25*b^7*d*e
^7 + 343*a*b^6*e^8)*m^5 + 245*(3*b^7*d*e^7 + 61*a*b^6*e^8)*m^4 + 56*(29*b^7*d*e^7 + 938*a*b^6*e^8)*m^3 + 196*(
9*b^7*d*e^7 + 527*a*b^6*e^8)*m^2 + 144*(5*b^7*d*e^7 + 721*a*b^6*e^8)*m)*x^7 - 7*(a^6*b*d^2*e^6 - 5*a^7*d*e^7)*
m^6 + 7*(20160*a^2*b^5*e^8 + (a*b^6*d*e^7 + 3*a^2*b^5*e^8)*m^7 - (b^7*d^2*e^6 - 23*a*b^6*d*e^7 - 90*a^2*b^5*e^
8)*m^6 - (15*b^7*d^2*e^6 - 205*a*b^6*d*e^7 - 1098*a^2*b^5*e^8)*m^5 - 5*(17*b^7*d^2*e^6 - 181*a*b^6*d*e^7 - 140
4*a^2*b^5*e^8)*m^4 - (225*b^7*d^2*e^6 - 2074*a*b^6*d*e^7 - 25227*a^2*b^5*e^8)*m^3 - 2*(137*b^7*d^2*e^6 - 1156*
a*b^6*d*e^7 - 25245*a^2*b^5*e^8)*m^2 - 24*(5*b^7*d^2*e^6 - 40*a*b^6*d*e^7 - 2143*a^2*b^5*e^8)*m)*x^6 + 7*(6*a^
5*b^2*d^3*e^5 - 33*a^6*b*d^2*e^6 + 73*a^7*d*e^7)*m^5 + 7*(40320*a^3*b^4*e^8 + (3*a^2*b^5*d*e^7 + 5*a^3*b^4*e^8
)*m^7 - (6*a*b^6*d^2*e^6 - 75*a^2*b^5*d*e^7 - 155*a^3*b^4*e^8)*m^6 + (6*b^7*d^3*e^5 - 108*a*b^6*d^2*e^6 + 723*
a^2*b^5*d*e^7 + 1955*a^3*b^4*e^8)*m^5 + 5*(12*b^7*d^3*e^5 - 138*a*b^6*d^2*e^6 + 681*a^2*b^5*d*e^7 + 2581*a^3*b
^4*e^8)*m^4 + 2*(105*b^7*d^3*e^5 - 990*a*b^6*d^2*e^6 + 4101*a^2*b^5*d*e^7 + 23860*a^3*b^4*e^8)*m^3 + 4*(75*b^7
*d^3*e^5 - 636*a*b^6*d^2*e^6 + 2370*a^2*b^5*d*e^7 + 24455*a^3*b^4*e^8)*m^2 + 144*(b^7*d^3*e^5 - 8*a*b^6*d^2*e^
6 + 28*a^2*b^5*d*e^7 + 705*a^3*b^4*e^8)*m)*x^5 - 35*(6*a^4*b^3*d^4*e^4 - 36*a^5*b^2*d^3*e^5 + 89*a^6*b*d^2*e^6
- 115*a^7*d*e^7)*m^4 + 35*(10080*a^4*b^3*e^8 + (a^3*b^4*d*e^7 + a^4*b^3*e^8)*m^7 - (3*a^2*b^5*d^2*e^6 - 27*a^
3*b^4*d*e^7 - 32*a^4*b^3*e^8)*m^6 + (6*a*b^6*d^3*e^5 - 63*a^2*b^5*d^2*e^6 + 283*a^3*b^4*d*e^7 + 418*a^4*b^3*e^
8)*m^5 - (6*b^7*d^4*e^4 - 84*a*b^6*d^3*e^5 + 471*a^2*b^5*d^2*e^6 - 1449*a^3*b^4*d*e^7 - 2864*a^4*b^3*e^8)*m^4
- (36*b^7*d^4*e^4 - 354*a*b^6*d^3*e^5 + 1521*a^2*b^5*d^2*e^6 - 3748*a^3*b^4*d*e^7 - 10993*a^4*b^3*e^8)*m^3 - 2
*(33*b^7*d^4*e^4 - 282*a*b^6*d^3*e^5 + 1059*a^2*b^5*d^2*e^6 - 2286*a^3*b^4*d*e^7 - 11656*a^4*b^3*e^8)*m^2 - 36
*(b^7*d^4*e^4 - 8*a*b^6*d^3*e^5 + 28*a^2*b^5*d^2*e^6 - 56*a^3*b^4*d*e^7 - 691*a^4*b^3*e^8)*m)*x^4 + 7*(120*a^3
*b^4*d^5*e^3 - 780*a^4*b^3*d^4*e^4 + 2130*a^5*b^2*d^3*e^5 - 3135*a^6*b*d^2*e^6 + 2632*a^7*d*e^7)*m^3 + 7*(4032
0*a^5*b^2*e^8 + (5*a^4*b^3*d*e^7 + 3*a^5*b^2*e^8)*m^7 - (20*a^3*b^4*d^2*e^6 - 145*a^4*b^3*d*e^7 - 99*a^5*b^2*e
^8)*m^6 + (60*a^2*b^5*d^3*e^5 - 480*a^3*b^4*d^2*e^6 + 1655*a^4*b^3*d*e^7 + 1341*a^5*b^2*e^8)*m^5 - 5*(24*a*b^6
*d^4*e^4 - 216*a^2*b^5*d^3*e^5 + 844*a^3*b^4*d^2*e^6 - 1871*a^4*b^3*d*e^7 - 1917*a^5*b^2*e^8)*m^4 + 4*(30*b^7*
d^5*e^3 - 330*a*b^6*d^4*e^4 + 1545*a^2*b^5*d^3*e^5 - 4080*a^3*b^4*d^2*e^6 + 6725*a^4*b^3*d*e^7 + 9648*a^5*b^2*
e^8)*m^3 + 4*(90*b^7*d^5*e^3 - 780*a*b^6*d^4*e^4 + 2970*a^2*b^5*d^3*e^5 - 6500*a^3*b^4*d^2*e^6 + 8965*a^4*b^3*
d*e^7 + 21519*a^5*b^2*e^8)*m^2 + 48*(5*b^7*d^5*e^3 - 40*a*b^6*d^4*e^4 + 140*a^2*b^5*d^3*e^5 - 280*a^3*b^4*d^2*
e^6 + 350*a^4*b^3*d*e^7 + 2003*a^5*b^2*e^8)*m)*x^3 - 14*(180*a^2*b^5*d^6*e^2 - 1260*a^3*b^4*d^5*e^3 + 3765*a^4
*b^3*d^4*e^4 - 6210*a^5*b^2*d^3*e^5 + 6077*a^6*b*d^2*e^6 - 3490*a^7*d*e^7)*m^2 + 7*(20160*a^6*b*e^8 + (3*a^5*b
^2*d*e^7 + a^6*b*e^8)*m^7 - (15*a^4*b^3*d^2*e^6 - 93*a^5*b^2*d*e^7 - 34*a^6*b*e^8)*m^6 + (60*a^3*b^4*d^3*e^5 -
405*a^4*b^3*d^2*e^6 + 1155*a^5*b^2*d*e^7 + 478*a^6*b*e^8)*m^5 - 5*(36*a^2*b^5*d^4*e^4 - 264*a^3*b^4*d^3*e^5 +
831*a^4*b^3*d^2*e^6 - 1455*a^5*b^2*d*e^7 - 716*a^6*b*e^8)*m^4 + (360*a*b^6*d^5*e^3 - 2880*a^2*b^5*d^4*e^4 + 1
0020*a^3*b^4*d^3*e^5 - 19755*a^4*b^3*d^2*e^6 + 24042*a^5*b^2*d*e^7 + 15289*a^6*b*e^8)*m^3 - 2*(180*b^7*d^6*e^2
- 1620*a*b^6*d^5*e^3 + 6390*a^2*b^5*d^4*e^4 - 14460*a^3*b^4*d^3*e^5 + 20595*a^4*b^3*d^2*e^6 - 18996*a^5*b^2*d
*e^7 - 18353*a^6*b*e^8)*m^2 - 72*(5*b^7*d^6*e^2 - 40*a*b^6*d^5*e^3 + 140*a^2*b^5*d^4*e^4 - 280*a^3*b^4*d^3*e^5
+ 350*a^4*b^3*d^2*e^6 - 280*a^5*b^2*d*e^7 - 621*a^6*b*e^8)*m)*x^2 + 12*(420*a*b^6*d^7*e - 3150*a^2*b^5*d^6*e^
2 + 10220*a^3*b^4*d^5*e^3 - 18655*a^4*b^3*d^4*e^4 + 20804*a^5*b^2*d^3*e^5 - 14322*a^6*b*d^2*e^6 + 5772*a^7*d*e
^7)*m + (40320*a^7*e^8 + (7*a^6*b*d*e^7 + a^7*e^8)*m^7 - 7*(6*a^5*b^2*d^2*e^6 - 33*a^6*b*d*e^7 - 5*a^7*e^8)*m^
6 + 7*(30*a^4*b^3*d^3*e^5 - 180*a^5*b^2*d^2*e^6 + 445*a^6*b*d*e^7 + 73*a^7*e^8)*m^5 - 35*(24*a^3*b^4*d^4*e^4 -
156*a^4*b^3*d^3*e^5 + 426*a^5*b^2*d^2*e^6 - 627*a^6*b*d*e^7 - 115*a^7*e^8)*m^4 + 14*(180*a^2*b^5*d^5*e^3 - 12
60*a^3*b^4*d^4*e^4 + 3765*a^4*b^3*d^3*e^5 - 6210*a^5*b^2*d^2*e^6 + 6077*a^6*b*d*e^7 + 1316*a^7*e^8)*m^3 - 28*(
180*a*b^6*d^6*e^2 - 1350*a^2*b^5*d^5*e^3 + 4380*a^3*b^4*d^4*e^4 - 7995*a^4*b^3*d^3*e^5 + 8916*a^5*b^2*d^2*e^6
- 6138*a^6*b*d*e^7 - 1745*a^7*e^8)*m^2 + 144*(35*b^7*d^7*e - 280*a*b^6*d^6*e^2 + 980*a^2*b^5*d^5*e^3 - 1960*a^
3*b^4*d^4*e^4 + 2450*a^4*b^3*d^3*e^5 - 1960*a^5*b^2*d^2*e^6 + 980*a^6*b*d*e^7 + 481*a^7*e^8)*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)

________________________________________________________________________________________

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*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.28407, size = 7613, normalized size = 31.85 \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*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

((x*e + d)^m*b^7*m^7*x^8*e^8 + (x*e + d)^m*b^7*d*m^7*x^7*e^7 + 7*(x*e + d)^m*a*b^6*m^7*x^7*e^8 + 28*(x*e + d)^
m*b^7*m^6*x^8*e^8 + 7*(x*e + d)^m*a*b^6*d*m^7*x^6*e^7 + 21*(x*e + d)^m*b^7*d*m^6*x^7*e^7 - 7*(x*e + d)^m*b^7*d
^2*m^6*x^6*e^6 + 21*(x*e + d)^m*a^2*b^5*m^7*x^6*e^8 + 203*(x*e + d)^m*a*b^6*m^6*x^7*e^8 + 322*(x*e + d)^m*b^7*
m^5*x^8*e^8 + 21*(x*e + d)^m*a^2*b^5*d*m^7*x^5*e^7 + 161*(x*e + d)^m*a*b^6*d*m^6*x^6*e^7 + 175*(x*e + d)^m*b^7
*d*m^5*x^7*e^7 - 42*(x*e + d)^m*a*b^6*d^2*m^6*x^5*e^6 - 105*(x*e + d)^m*b^7*d^2*m^5*x^6*e^6 + 42*(x*e + d)^m*b
^7*d^3*m^5*x^5*e^5 + 35*(x*e + d)^m*a^3*b^4*m^7*x^5*e^8 + 630*(x*e + d)^m*a^2*b^5*m^6*x^6*e^8 + 2401*(x*e + d)
^m*a*b^6*m^5*x^7*e^8 + 1960*(x*e + d)^m*b^7*m^4*x^8*e^8 + 35*(x*e + d)^m*a^3*b^4*d*m^7*x^4*e^7 + 525*(x*e + d)
^m*a^2*b^5*d*m^6*x^5*e^7 + 1435*(x*e + d)^m*a*b^6*d*m^5*x^6*e^7 + 735*(x*e + d)^m*b^7*d*m^4*x^7*e^7 - 105*(x*e
+ d)^m*a^2*b^5*d^2*m^6*x^4*e^6 - 756*(x*e + d)^m*a*b^6*d^2*m^5*x^5*e^6 - 595*(x*e + d)^m*b^7*d^2*m^4*x^6*e^6
+ 210*(x*e + d)^m*a*b^6*d^3*m^5*x^4*e^5 + 420*(x*e + d)^m*b^7*d^3*m^4*x^5*e^5 - 210*(x*e + d)^m*b^7*d^4*m^4*x^
4*e^4 + 35*(x*e + d)^m*a^4*b^3*m^7*x^4*e^8 + 1085*(x*e + d)^m*a^3*b^4*m^6*x^5*e^8 + 7686*(x*e + d)^m*a^2*b^5*m
^5*x^6*e^8 + 14945*(x*e + d)^m*a*b^6*m^4*x^7*e^8 + 6769*(x*e + d)^m*b^7*m^3*x^8*e^8 + 35*(x*e + d)^m*a^4*b^3*d
*m^7*x^3*e^7 + 945*(x*e + d)^m*a^3*b^4*d*m^6*x^4*e^7 + 5061*(x*e + d)^m*a^2*b^5*d*m^5*x^5*e^7 + 6335*(x*e + d)
^m*a*b^6*d*m^4*x^6*e^7 + 1624*(x*e + d)^m*b^7*d*m^3*x^7*e^7 - 140*(x*e + d)^m*a^3*b^4*d^2*m^6*x^3*e^6 - 2205*(
x*e + d)^m*a^2*b^5*d^2*m^5*x^4*e^6 - 4830*(x*e + d)^m*a*b^6*d^2*m^4*x^5*e^6 - 1575*(x*e + d)^m*b^7*d^2*m^3*x^6
*e^6 + 420*(x*e + d)^m*a^2*b^5*d^3*m^5*x^3*e^5 + 2940*(x*e + d)^m*a*b^6*d^3*m^4*x^4*e^5 + 1470*(x*e + d)^m*b^7
*d^3*m^3*x^5*e^5 - 840*(x*e + d)^m*a*b^6*d^4*m^4*x^3*e^4 - 1260*(x*e + d)^m*b^7*d^4*m^3*x^4*e^4 + 840*(x*e + d
)^m*b^7*d^5*m^3*x^3*e^3 + 21*(x*e + d)^m*a^5*b^2*m^7*x^3*e^8 + 1120*(x*e + d)^m*a^4*b^3*m^6*x^4*e^8 + 13685*(x
*e + d)^m*a^3*b^4*m^5*x^5*e^8 + 49140*(x*e + d)^m*a^2*b^5*m^4*x^6*e^8 + 52528*(x*e + d)^m*a*b^6*m^3*x^7*e^8 +
13132*(x*e + d)^m*b^7*m^2*x^8*e^8 + 21*(x*e + d)^m*a^5*b^2*d*m^7*x^2*e^7 + 1015*(x*e + d)^m*a^4*b^3*d*m^6*x^3*
e^7 + 9905*(x*e + d)^m*a^3*b^4*d*m^5*x^4*e^7 + 23835*(x*e + d)^m*a^2*b^5*d*m^4*x^5*e^7 + 14518*(x*e + d)^m*a*b
^6*d*m^3*x^6*e^7 + 1764*(x*e + d)^m*b^7*d*m^2*x^7*e^7 - 105*(x*e + d)^m*a^4*b^3*d^2*m^6*x^2*e^6 - 3360*(x*e +
d)^m*a^3*b^4*d^2*m^5*x^3*e^6 - 16485*(x*e + d)^m*a^2*b^5*d^2*m^4*x^4*e^6 - 13860*(x*e + d)^m*a*b^6*d^2*m^3*x^5
*e^6 - 1918*(x*e + d)^m*b^7*d^2*m^2*x^6*e^6 + 420*(x*e + d)^m*a^3*b^4*d^3*m^5*x^2*e^5 + 7560*(x*e + d)^m*a^2*b
^5*d^3*m^4*x^3*e^5 + 12390*(x*e + d)^m*a*b^6*d^3*m^3*x^4*e^5 + 2100*(x*e + d)^m*b^7*d^3*m^2*x^5*e^5 - 1260*(x*
e + d)^m*a^2*b^5*d^4*m^4*x^2*e^4 - 9240*(x*e + d)^m*a*b^6*d^4*m^3*x^3*e^4 - 2310*(x*e + d)^m*b^7*d^4*m^2*x^4*e
^4 + 2520*(x*e + d)^m*a*b^6*d^5*m^3*x^2*e^3 + 2520*(x*e + d)^m*b^7*d^5*m^2*x^3*e^3 - 2520*(x*e + d)^m*b^7*d^6*
m^2*x^2*e^2 + 7*(x*e + d)^m*a^6*b*m^7*x^2*e^8 + 693*(x*e + d)^m*a^5*b^2*m^6*x^3*e^8 + 14630*(x*e + d)^m*a^4*b^
3*m^5*x^4*e^8 + 90335*(x*e + d)^m*a^3*b^4*m^4*x^5*e^8 + 176589*(x*e + d)^m*a^2*b^5*m^3*x^6*e^8 + 103292*(x*e +
d)^m*a*b^6*m^2*x^7*e^8 + 13068*(x*e + d)^m*b^7*m*x^8*e^8 + 7*(x*e + d)^m*a^6*b*d*m^7*x*e^7 + 651*(x*e + d)^m*
a^5*b^2*d*m^6*x^2*e^7 + 11585*(x*e + d)^m*a^4*b^3*d*m^5*x^3*e^7 + 50715*(x*e + d)^m*a^3*b^4*d*m^4*x^4*e^7 + 57
414*(x*e + d)^m*a^2*b^5*d*m^3*x^5*e^7 + 16184*(x*e + d)^m*a*b^6*d*m^2*x^6*e^7 + 720*(x*e + d)^m*b^7*d*m*x^7*e^
7 - 42*(x*e + d)^m*a^5*b^2*d^2*m^6*x*e^6 - 2835*(x*e + d)^m*a^4*b^3*d^2*m^5*x^2*e^6 - 29540*(x*e + d)^m*a^3*b^
4*d^2*m^4*x^3*e^6 - 53235*(x*e + d)^m*a^2*b^5*d^2*m^3*x^4*e^6 - 17808*(x*e + d)^m*a*b^6*d^2*m^2*x^5*e^6 - 840*
(x*e + d)^m*b^7*d^2*m*x^6*e^6 + 210*(x*e + d)^m*a^4*b^3*d^3*m^5*x*e^5 + 9240*(x*e + d)^m*a^3*b^4*d^3*m^4*x^2*e
^5 + 43260*(x*e + d)^m*a^2*b^5*d^3*m^3*x^3*e^5 + 19740*(x*e + d)^m*a*b^6*d^3*m^2*x^4*e^5 + 1008*(x*e + d)^m*b^
7*d^3*m*x^5*e^5 - 840*(x*e + d)^m*a^3*b^4*d^4*m^4*x*e^4 - 20160*(x*e + d)^m*a^2*b^5*d^4*m^3*x^2*e^4 - 21840*(x
*e + d)^m*a*b^6*d^4*m^2*x^3*e^4 - 1260*(x*e + d)^m*b^7*d^4*m*x^4*e^4 + 2520*(x*e + d)^m*a^2*b^5*d^5*m^3*x*e^3
+ 22680*(x*e + d)^m*a*b^6*d^5*m^2*x^2*e^3 + 1680*(x*e + d)^m*b^7*d^5*m*x^3*e^3 - 5040*(x*e + d)^m*a*b^6*d^6*m^
2*x*e^2 - 2520*(x*e + d)^m*b^7*d^6*m*x^2*e^2 + 5040*(x*e + d)^m*b^7*d^7*m*x*e + (x*e + d)^m*a^7*m^7*x*e^8 + 23
8*(x*e + d)^m*a^6*b*m^6*x^2*e^8 + 9387*(x*e + d)^m*a^5*b^2*m^5*x^3*e^8 + 100240*(x*e + d)^m*a^4*b^3*m^4*x^4*e^
8 + 334040*(x*e + d)^m*a^3*b^4*m^3*x^5*e^8 + 353430*(x*e + d)^m*a^2*b^5*m^2*x^6*e^8 + 103824*(x*e + d)^m*a*b^6
*m*x^7*e^8 + 5040*(x*e + d)^m*b^7*x^8*e^8 + (x*e + d)^m*a^7*d*m^7*e^7 + 231*(x*e + d)^m*a^6*b*d*m^6*x*e^7 + 80
85*(x*e + d)^m*a^5*b^2*d*m^5*x^2*e^7 + 65485*(x*e + d)^m*a^4*b^3*d*m^4*x^3*e^7 + 131180*(x*e + d)^m*a^3*b^4*d*
m^3*x^4*e^7 + 66360*(x*e + d)^m*a^2*b^5*d*m^2*x^5*e^7 + 6720*(x*e + d)^m*a*b^6*d*m*x^6*e^7 - 7*(x*e + d)^m*a^6
*b*d^2*m^6*e^6 - 1260*(x*e + d)^m*a^5*b^2*d^2*m^5*x*e^6 - 29085*(x*e + d)^m*a^4*b^3*d^2*m^4*x^2*e^6 - 114240*(
x*e + d)^m*a^3*b^4*d^2*m^3*x^3*e^6 - 74130*(x*e + d)^m*a^2*b^5*d^2*m^2*x^4*e^6 - 8064*(x*e + d)^m*a*b^6*d^2*m*
x^5*e^6 + 42*(x*e + d)^m*a^5*b^2*d^3*m^5*e^5 + 5460*(x*e + d)^m*a^4*b^3*d^3*m^4*x*e^5 + 70140*(x*e + d)^m*a^3*
b^4*d^3*m^3*x^2*e^5 + 83160*(x*e + d)^m*a^2*b^5*d^3*m^2*x^3*e^5 + 10080*(x*e + d)^m*a*b^6*d^3*m*x^4*e^5 - 210*
(x*e + d)^m*a^4*b^3*d^4*m^4*e^4 - 17640*(x*e + d)^m*a^3*b^4*d^4*m^3*x*e^4 - 89460*(x*e + d)^m*a^2*b^5*d^4*m^2*
x^2*e^4 - 13440*(x*e + d)^m*a*b^6*d^4*m*x^3*e^4 + 840*(x*e + d)^m*a^3*b^4*d^5*m^3*e^3 + 37800*(x*e + d)^m*a^2*
b^5*d^5*m^2*x*e^3 + 20160*(x*e + d)^m*a*b^6*d^5*m*x^2*e^3 - 2520*(x*e + d)^m*a^2*b^5*d^6*m^2*e^2 - 40320*(x*e
+ d)^m*a*b^6*d^6*m*x*e^2 + 5040*(x*e + d)^m*a*b^6*d^7*m*e - 5040*(x*e + d)^m*b^7*d^8 + 35*(x*e + d)^m*a^7*m^6*
x*e^8 + 3346*(x*e + d)^m*a^6*b*m^5*x^2*e^8 + 67095*(x*e + d)^m*a^5*b^2*m^4*x^3*e^8 + 384755*(x*e + d)^m*a^4*b^
3*m^3*x^4*e^8 + 684740*(x*e + d)^m*a^3*b^4*m^2*x^5*e^8 + 360024*(x*e + d)^m*a^2*b^5*m*x^6*e^8 + 40320*(x*e + d
)^m*a*b^6*x^7*e^8 + 35*(x*e + d)^m*a^7*d*m^6*e^7 + 3115*(x*e + d)^m*a^6*b*d*m^5*x*e^7 + 50925*(x*e + d)^m*a^5*
b^2*d*m^4*x^2*e^7 + 188300*(x*e + d)^m*a^4*b^3*d*m^3*x^3*e^7 + 160020*(x*e + d)^m*a^3*b^4*d*m^2*x^4*e^7 + 2822
4*(x*e + d)^m*a^2*b^5*d*m*x^5*e^7 - 231*(x*e + d)^m*a^6*b*d^2*m^5*e^6 - 14910*(x*e + d)^m*a^5*b^2*d^2*m^4*x*e^
6 - 138285*(x*e + d)^m*a^4*b^3*d^2*m^3*x^2*e^6 - 182000*(x*e + d)^m*a^3*b^4*d^2*m^2*x^3*e^6 - 35280*(x*e + d)^
m*a^2*b^5*d^2*m*x^4*e^6 + 1260*(x*e + d)^m*a^5*b^2*d^3*m^4*e^5 + 52710*(x*e + d)^m*a^4*b^3*d^3*m^3*x*e^5 + 202
440*(x*e + d)^m*a^3*b^4*d^3*m^2*x^2*e^5 + 47040*(x*e + d)^m*a^2*b^5*d^3*m*x^3*e^5 - 5460*(x*e + d)^m*a^4*b^3*d
^4*m^3*e^4 - 122640*(x*e + d)^m*a^3*b^4*d^4*m^2*x*e^4 - 70560*(x*e + d)^m*a^2*b^5*d^4*m*x^2*e^4 + 17640*(x*e +
d)^m*a^3*b^4*d^5*m^2*e^3 + 141120*(x*e + d)^m*a^2*b^5*d^5*m*x*e^3 - 37800*(x*e + d)^m*a^2*b^5*d^6*m*e^2 + 403
20*(x*e + d)^m*a*b^6*d^7*e + 511*(x*e + d)^m*a^7*m^5*x*e^8 + 25060*(x*e + d)^m*a^6*b*m^4*x^2*e^8 + 270144*(x*e
+ d)^m*a^5*b^2*m^3*x^3*e^8 + 815920*(x*e + d)^m*a^4*b^3*m^2*x^4*e^8 + 710640*(x*e + d)^m*a^3*b^4*m*x^5*e^8 +
141120*(x*e + d)^m*a^2*b^5*x^6*e^8 + 511*(x*e + d)^m*a^7*d*m^5*e^7 + 21945*(x*e + d)^m*a^6*b*d*m^4*x*e^7 + 168
294*(x*e + d)^m*a^5*b^2*d*m^3*x^2*e^7 + 251020*(x*e + d)^m*a^4*b^3*d*m^2*x^3*e^7 + 70560*(x*e + d)^m*a^3*b^4*d
*m*x^4*e^7 - 3115*(x*e + d)^m*a^6*b*d^2*m^4*e^6 - 86940*(x*e + d)^m*a^5*b^2*d^2*m^3*x*e^6 - 288330*(x*e + d)^m
*a^4*b^3*d^2*m^2*x^2*e^6 - 94080*(x*e + d)^m*a^3*b^4*d^2*m*x^3*e^6 + 14910*(x*e + d)^m*a^5*b^2*d^3*m^3*e^5 + 2
23860*(x*e + d)^m*a^4*b^3*d^3*m^2*x*e^5 + 141120*(x*e + d)^m*a^3*b^4*d^3*m*x^2*e^5 - 52710*(x*e + d)^m*a^4*b^3
*d^4*m^2*e^4 - 282240*(x*e + d)^m*a^3*b^4*d^4*m*x*e^4 + 122640*(x*e + d)^m*a^3*b^4*d^5*m*e^3 - 141120*(x*e + d
)^m*a^2*b^5*d^6*e^2 + 4025*(x*e + d)^m*a^7*m^4*x*e^8 + 107023*(x*e + d)^m*a^6*b*m^3*x^2*e^8 + 602532*(x*e + d)
^m*a^5*b^2*m^2*x^3*e^8 + 870660*(x*e + d)^m*a^4*b^3*m*x^4*e^8 + 282240*(x*e + d)^m*a^3*b^4*x^5*e^8 + 4025*(x*e
+ d)^m*a^7*d*m^4*e^7 + 85078*(x*e + d)^m*a^6*b*d*m^3*x*e^7 + 265944*(x*e + d)^m*a^5*b^2*d*m^2*x^2*e^7 + 11760
0*(x*e + d)^m*a^4*b^3*d*m*x^3*e^7 - 21945*(x*e + d)^m*a^6*b*d^2*m^3*e^6 - 249648*(x*e + d)^m*a^5*b^2*d^2*m^2*x
*e^6 - 176400*(x*e + d)^m*a^4*b^3*d^2*m*x^2*e^6 + 86940*(x*e + d)^m*a^5*b^2*d^3*m^2*e^5 + 352800*(x*e + d)^m*a
^4*b^3*d^3*m*x*e^5 - 223860*(x*e + d)^m*a^4*b^3*d^4*m*e^4 + 282240*(x*e + d)^m*a^3*b^4*d^5*e^3 + 18424*(x*e +
d)^m*a^7*m^3*x*e^8 + 256942*(x*e + d)^m*a^6*b*m^2*x^2*e^8 + 673008*(x*e + d)^m*a^5*b^2*m*x^3*e^8 + 352800*(x*e
+ d)^m*a^4*b^3*x^4*e^8 + 18424*(x*e + d)^m*a^7*d*m^3*e^7 + 171864*(x*e + d)^m*a^6*b*d*m^2*x*e^7 + 141120*(x*e
+ d)^m*a^5*b^2*d*m*x^2*e^7 - 85078*(x*e + d)^m*a^6*b*d^2*m^2*e^6 - 282240*(x*e + d)^m*a^5*b^2*d^2*m*x*e^6 + 2
49648*(x*e + d)^m*a^5*b^2*d^3*m*e^5 - 352800*(x*e + d)^m*a^4*b^3*d^4*e^4 + 48860*(x*e + d)^m*a^7*m^2*x*e^8 + 3
12984*(x*e + d)^m*a^6*b*m*x^2*e^8 + 282240*(x*e + d)^m*a^5*b^2*x^3*e^8 + 48860*(x*e + d)^m*a^7*d*m^2*e^7 + 141
120*(x*e + d)^m*a^6*b*d*m*x*e^7 - 171864*(x*e + d)^m*a^6*b*d^2*m*e^6 + 282240*(x*e + d)^m*a^5*b^2*d^3*e^5 + 69
264*(x*e + d)^m*a^7*m*x*e^8 + 141120*(x*e + d)^m*a^6*b*x^2*e^8 + 69264*(x*e + d)^m*a^7*d*m*e^7 - 141120*(x*e +
d)^m*a^6*b*d^2*e^6 + 40320*(x*e + d)^m*a^7*x*e^8 + 40320*(x*e + d)^m*a^7*d*e^7)/(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 + 109584*m*e^8 + 40320*e^8)