### 3.2550 $$\int (d+e x)^m (a+b x+c x^2)^3 \, dx$$

Optimal. Leaf size=305 $\frac{3 (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (m+3)}-\frac{(2 c d-b e) (d+e x)^{m+4} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (m+4)}+\frac{3 c (d+e x)^{m+5} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (m+5)}+\frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^7 (m+1)}-\frac{3 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2}{e^7 (m+2)}-\frac{3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)}$

[Out]

((c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d +
e*x)^(2 + m))/(e^7*(2 + m)) + (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(
3 + m))/(e^7*(3 + m)) - ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(4 + m))/(e^7*
(4 + m)) + (3*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (3*c^2*(2*c*d - b
*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)^(7 + m))/(e^7*(7 + m))

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Rubi [A]  time = 0.211511, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {698} $\frac{3 (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (m+3)}-\frac{(2 c d-b e) (d+e x)^{m+4} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (m+4)}+\frac{3 c (d+e x)^{m+5} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (m+5)}+\frac{(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^7 (m+1)}-\frac{3 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2}{e^7 (m+2)}-\frac{3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac{c^3 (d+e x)^{m+7}}{e^7 (m+7)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d +
e*x)^(2 + m))/(e^7*(2 + m)) + (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(
3 + m))/(e^7*(3 + m)) - ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(4 + m))/(e^7*
(4 + m)) + (3*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (3*c^2*(2*c*d - b
*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)^(7 + m))/(e^7*(7 + m))

Rule 698

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

Rubi steps

\begin{align*} \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^m}{e^6}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{1+m}}{e^6}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{2+m}}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{3+m}}{e^6}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{4+m}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{5+m}}{e^6}+\frac{c^3 (d+e x)^{6+m}}{e^6}\right ) \, dx\\ &=\frac{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{2+m}}{e^7 (2+m)}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac{3 c^2 (2 c d-b e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac{c^3 (d+e x)^{7+m}}{e^7 (7+m)}\\ \end{align*}

Mathematica [A]  time = 1.70416, size = 476, normalized size = 1.56 $\frac{(d+e x)^{m+1} \left (\frac{3 (d+e x) \left (-2 (m+2) (d+e x) \left (2 c^2 e^2 \left (4 a^2 e^2 \left (m^2+10 m+24\right )+4 a b d e \left (m^2-2 m-33\right )+b^2 d^2 \left (m^2-2 m+57\right )\right )-2 b^2 c e^3 (m+1) (3 a e (m+5)+b d (m-3))-8 c^3 d^2 e \left (a e \left (m^2-2 m-33\right )+30 b d\right )+b^4 e^4 \left (m^2+4 m+3\right )+120 c^4 d^4\right )-2 e^2 (m+2) (m+3) (a+x (b+c x)) \left (2 c^2 e \left (2 a e \left (d \left (m^2+3 m-13\right )+e \left (m^2+10 m+24\right ) x\right )+5 b d (d (m+13)-2 e (m+4) x)\right )-b c e^2 \left (-2 a e (7 m+37)+b d \left (m^2+13 m+72\right )+b e \left (m^2+5 m+4\right ) x\right )-b^3 e^3 (m+1)+20 c^3 d^2 (e (m+4) x-3 d)\right )+2 (m+3) (2 c d-b e) \left (e (a e-b d)+c d^2\right ) \left (-4 c e \left (a e \left (m^2+3 m-13\right )+15 b d\right )+b^2 e^2 \left (m^2+3 m+2\right )+60 c^2 d^2\right )-c e^4 (m+2) (m+3) (m+4) (m+5) (a+x (b+c x))^2 (b e (m+11)+2 c (e (m+6) x-5 d))\right )}{c e^6 (m+2) (m+3) (m+4) (m+5) (m+6) (m+7)}+(a+x (b+c x))^3\right )}{e (m+1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((d + e*x)^(1 + m)*((a + x*(b + c*x))^3 + (3*(d + e*x)*(2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(3 + m)*(60
*c^2*d^2 + b^2*e^2*(2 + 3*m + m^2) - 4*c*e*(15*b*d + a*e*(-13 + 3*m + m^2))) - 2*(2 + m)*(120*c^4*d^4 + b^4*e^
4*(3 + 4*m + m^2) - 2*b^2*c*e^3*(1 + m)*(b*d*(-3 + m) + 3*a*e*(5 + m)) - 8*c^3*d^2*e*(30*b*d + a*e*(-33 - 2*m
+ m^2)) + 2*c^2*e^2*(4*a*b*d*e*(-33 - 2*m + m^2) + b^2*d^2*(57 - 2*m + m^2) + 4*a^2*e^2*(24 + 10*m + m^2)))*(d
+ e*x) - c*e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + x*(b + c*x))^2*(b*e*(11 + m) + 2*c*(-5*d + e*(6 + m)*x))
- 2*e^2*(2 + m)*(3 + m)*(a + x*(b + c*x))*(-(b^3*e^3*(1 + m)) + 20*c^3*d^2*(-3*d + e*(4 + m)*x) - b*c*e^2*(-2*
a*e*(37 + 7*m) + b*d*(72 + 13*m + m^2) + b*e*(4 + 5*m + m^2)*x) + 2*c^2*e*(5*b*d*(d*(13 + m) - 2*e*(4 + m)*x)
+ 2*a*e*(d*(-13 + 3*m + m^2) + e*(24 + 10*m + m^2)*x)))))/(c*e^6*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 +
m))))/(e*(1 + m))

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Maple [B]  time = 0.059, size = 2927, normalized size = 9.6 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(c*x^2+b*x+a)^3,x)

[Out]

(e*x+d)^(1+m)*(c^3*e^6*m^6*x^6+3*b*c^2*e^6*m^6*x^5+21*c^3*e^6*m^5*x^6+3*a*c^2*e^6*m^6*x^4+3*b^2*c*e^6*m^6*x^4+
66*b*c^2*e^6*m^5*x^5-6*c^3*d*e^5*m^5*x^5+175*c^3*e^6*m^4*x^6+6*a*b*c*e^6*m^6*x^3+69*a*c^2*e^6*m^5*x^4+b^3*e^6*
m^6*x^3+69*b^2*c*e^6*m^5*x^4-15*b*c^2*d*e^5*m^5*x^4+570*b*c^2*e^6*m^4*x^5-90*c^3*d*e^5*m^4*x^5+735*c^3*e^6*m^3
*x^6+3*a^2*c*e^6*m^6*x^2+3*a*b^2*e^6*m^6*x^2+144*a*b*c*e^6*m^5*x^3-12*a*c^2*d*e^5*m^5*x^3+621*a*c^2*e^6*m^4*x^
4+24*b^3*e^6*m^5*x^3-12*b^2*c*d*e^5*m^5*x^3+621*b^2*c*e^6*m^4*x^4-255*b*c^2*d*e^5*m^4*x^4+2460*b*c^2*e^6*m^3*x
^5+30*c^3*d^2*e^4*m^4*x^4-510*c^3*d*e^5*m^3*x^5+1624*c^3*e^6*m^2*x^6+3*a^2*b*e^6*m^6*x+75*a^2*c*e^6*m^5*x^2+75
*a*b^2*e^6*m^5*x^2-18*a*b*c*d*e^5*m^5*x^2+1356*a*b*c*e^6*m^4*x^3-228*a*c^2*d*e^5*m^4*x^3+2775*a*c^2*e^6*m^3*x^
4-3*b^3*d*e^5*m^5*x^2+226*b^3*e^6*m^4*x^3-228*b^2*c*d*e^5*m^4*x^3+2775*b^2*c*e^6*m^3*x^4+60*b*c^2*d^2*e^4*m^4*
x^3-1575*b*c^2*d*e^5*m^3*x^4+5547*b*c^2*e^6*m^2*x^5+300*c^3*d^2*e^4*m^3*x^4-1350*c^3*d*e^5*m^2*x^5+1764*c^3*e^
6*m*x^6+a^3*e^6*m^6+78*a^2*b*e^6*m^5*x-6*a^2*c*d*e^5*m^5*x+741*a^2*c*e^6*m^4*x^2-6*a*b^2*d*e^5*m^5*x+741*a*b^2
*e^6*m^4*x^2-378*a*b*c*d*e^5*m^4*x^2+6336*a*b*c*e^6*m^3*x^3+36*a*c^2*d^2*e^4*m^4*x^2-1572*a*c^2*d*e^5*m^3*x^3+
6432*a*c^2*e^6*m^2*x^4-63*b^3*d*e^5*m^4*x^2+1056*b^3*e^6*m^3*x^3+36*b^2*c*d^2*e^4*m^4*x^2-1572*b^2*c*d*e^5*m^3
*x^3+6432*b^2*c*e^6*m^2*x^4+780*b*c^2*d^2*e^4*m^3*x^3-4425*b*c^2*d*e^5*m^2*x^4+6114*b*c^2*e^6*m*x^5-120*c^3*d^
3*e^3*m^3*x^3+1050*c^3*d^2*e^4*m^2*x^4-1644*c^3*d*e^5*m*x^5+720*c^3*e^6*x^6+27*a^3*e^6*m^5-3*a^2*b*d*e^5*m^5+8
10*a^2*b*e^6*m^4*x-138*a^2*c*d*e^5*m^4*x+3657*a^2*c*e^6*m^3*x^2-138*a*b^2*d*e^5*m^4*x+3657*a*b^2*e^6*m^3*x^2+3
6*a*b*c*d^2*e^4*m^4*x-2934*a*b*c*d*e^5*m^3*x^2+15270*a*b*c*e^6*m^2*x^3+576*a*c^2*d^2*e^4*m^3*x^2-4812*a*c^2*d*
e^5*m^2*x^3+7236*a*c^2*e^6*m*x^4+6*b^3*d^2*e^4*m^4*x-489*b^3*d*e^5*m^3*x^2+2545*b^3*e^6*m^2*x^3+576*b^2*c*d^2*
e^4*m^3*x^2-4812*b^2*c*d*e^5*m^2*x^3+7236*b^2*c*e^6*m*x^4-180*b*c^2*d^3*e^3*m^3*x^2+3180*b*c^2*d^2*e^4*m^2*x^3
-5610*b*c^2*d*e^5*m*x^4+2520*b*c^2*e^6*x^5-720*c^3*d^3*e^3*m^2*x^3+1500*c^3*d^2*e^4*m*x^4-720*c^3*d*e^5*x^5+29
5*a^3*e^6*m^4-75*a^2*b*d*e^5*m^4+4260*a^2*b*e^6*m^3*x+6*a^2*c*d^2*e^4*m^4-1206*a^2*c*d*e^5*m^3*x+9336*a^2*c*e^
6*m^2*x^2+6*a*b^2*d^2*e^4*m^4-1206*a*b^2*d*e^5*m^3*x+9336*a*b^2*e^6*m^2*x^2+684*a*b*c*d^2*e^4*m^3*x-10206*a*b*
c*d*e^5*m^2*x^2+17712*a*b*c*e^6*m*x^3-72*a*c^2*d^3*e^3*m^3*x+2988*a*c^2*d^2*e^4*m^2*x^2-6480*a*c^2*d*e^5*m*x^3
+3024*a*c^2*e^6*x^4+114*b^3*d^2*e^4*m^3*x-1701*b^3*d*e^5*m^2*x^2+2952*b^3*e^6*m*x^3-72*b^2*c*d^3*e^3*m^3*x+298
8*b^2*c*d^2*e^4*m^2*x^2-6480*b^2*c*d*e^5*m*x^3+3024*b^2*c*e^6*x^4-1800*b*c^2*d^3*e^3*m^2*x^2+4980*b*c^2*d^2*e^
4*m*x^3-2520*b*c^2*d*e^5*x^4+360*c^3*d^4*e^2*m^2*x^2-1320*c^3*d^3*e^3*m*x^3+720*c^3*d^2*e^4*x^4+1665*a^3*e^6*m
^3-735*a^2*b*d*e^5*m^3+11787*a^2*b*e^6*m^2*x+132*a^2*c*d^2*e^4*m^3-4902*a^2*c*d*e^5*m^2*x+11388*a^2*c*e^6*m*x^
2+132*a*b^2*d^2*e^4*m^3-4902*a*b^2*d*e^5*m^2*x+11388*a*b^2*e^6*m*x^2-36*a*b*c*d^3*e^3*m^3+4500*a*b*c*d^2*e^4*m
^2*x-15192*a*b*c*d*e^5*m*x^2+7560*a*b*c*e^6*x^3-1008*a*c^2*d^3*e^3*m^2*x+5472*a*c^2*d^2*e^4*m*x^2-3024*a*c^2*d
*e^5*x^3-6*b^3*d^3*e^3*m^3+750*b^3*d^2*e^4*m^2*x-2532*b^3*d*e^5*m*x^2+1260*b^3*e^6*x^3-1008*b^2*c*d^3*e^3*m^2*
x+5472*b^2*c*d^2*e^4*m*x^2-3024*b^2*c*d*e^5*x^3+360*b*c^2*d^4*e^2*m^2*x-4140*b*c^2*d^3*e^3*m*x^2+2520*b*c^2*d^
2*e^4*x^3+1080*c^3*d^4*e^2*m*x^2-720*c^3*d^3*e^3*x^3+5104*a^3*e^6*m^2-3525*a^2*b*d*e^5*m^2+15822*a^2*b*e^6*m*x
+1074*a^2*c*d^2*e^4*m^2-8868*a^2*c*d*e^5*m*x+5040*a^2*c*e^6*x^2+1074*a*b^2*d^2*e^4*m^2-8868*a*b^2*d*e^5*m*x+50
40*a*b^2*e^6*x^2-648*a*b*c*d^3*e^3*m^2+11412*a*b*c*d^2*e^4*m*x-7560*a*b*c*d*e^5*x^2+72*a*c^2*d^4*e^2*m^2-3960*
a*c^2*d^3*e^3*m*x+3024*a*c^2*d^2*e^4*x^2-108*b^3*d^3*e^3*m^2+1902*b^3*d^2*e^4*m*x-1260*b^3*d*e^5*x^2+72*b^2*c*
d^4*e^2*m^2-3960*b^2*c*d^3*e^3*m*x+3024*b^2*c*d^2*e^4*x^2+2880*b*c^2*d^4*e^2*m*x-2520*b*c^2*d^3*e^3*x^2-720*c^
3*d^5*e*m*x+720*c^3*d^4*e^2*x^2+8028*a^3*e^6*m-8262*a^2*b*d*e^5*m+7560*a^2*b*e^6*x+3828*a^2*c*d^2*e^4*m-5040*a
^2*c*d*e^5*x+3828*a*b^2*d^2*e^4*m-5040*a*b^2*d*e^5*x-3852*a*b*c*d^3*e^3*m+7560*a*b*c*d^2*e^4*x+936*a*c^2*d^4*e
^2*m-3024*a*c^2*d^3*e^3*x-642*b^3*d^3*e^3*m+1260*b^3*d^2*e^4*x+936*b^2*c*d^4*e^2*m-3024*b^2*c*d^3*e^3*x-360*b*
c^2*d^5*e*m+2520*b*c^2*d^4*e^2*x-720*c^3*d^5*e*x+5040*a^3*e^6-7560*a^2*b*d*e^5+5040*a^2*c*d^2*e^4+5040*a*b^2*d
^2*e^4-7560*a*b*c*d^3*e^3+3024*a*c^2*d^4*e^2-1260*b^3*d^3*e^3+3024*b^2*c*d^4*e^2-2520*b*c^2*d^5*e+720*c^3*d^6)
/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)

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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((e*x+d)^m*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.98024, size = 5573, normalized size = 18.27 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

(a^3*d*e^6*m^6 + 720*c^3*d^7 - 2520*b*c^2*d^6*e - 7560*a^2*b*d^2*e^5 + 5040*a^3*d*e^6 + 3024*(b^2*c + a*c^2)*d
^5*e^2 - 1260*(b^3 + 6*a*b*c)*d^4*e^3 + 5040*(a*b^2 + a^2*c)*d^3*e^4 + (c^3*e^7*m^6 + 21*c^3*e^7*m^5 + 175*c^3
*e^7*m^4 + 735*c^3*e^7*m^3 + 1624*c^3*e^7*m^2 + 1764*c^3*e^7*m + 720*c^3*e^7)*x^7 + (2520*b*c^2*e^7 + (c^3*d*e
^6 + 3*b*c^2*e^7)*m^6 + 3*(5*c^3*d*e^6 + 22*b*c^2*e^7)*m^5 + 5*(17*c^3*d*e^6 + 114*b*c^2*e^7)*m^4 + 15*(15*c^3
*d*e^6 + 164*b*c^2*e^7)*m^3 + (274*c^3*d*e^6 + 5547*b*c^2*e^7)*m^2 + 6*(20*c^3*d*e^6 + 1019*b*c^2*e^7)*m)*x^6
- 3*(a^2*b*d^2*e^5 - 9*a^3*d*e^6)*m^5 + 3*(1008*(b^2*c + a*c^2)*e^7 + (b*c^2*d*e^6 + (b^2*c + a*c^2)*e^7)*m^6
- (2*c^3*d^2*e^5 - 17*b*c^2*d*e^6 - 23*(b^2*c + a*c^2)*e^7)*m^5 - (20*c^3*d^2*e^5 - 105*b*c^2*d*e^6 - 207*(b^2
*c + a*c^2)*e^7)*m^4 - 5*(14*c^3*d^2*e^5 - 59*b*c^2*d*e^6 - 185*(b^2*c + a*c^2)*e^7)*m^3 - 2*(50*c^3*d^2*e^5 -
187*b*c^2*d*e^6 - 1072*(b^2*c + a*c^2)*e^7)*m^2 - 12*(4*c^3*d^2*e^5 - 14*b*c^2*d*e^6 - 201*(b^2*c + a*c^2)*e^
7)*m)*x^5 - (75*a^2*b*d^2*e^5 - 295*a^3*d*e^6 - 6*(a*b^2 + a^2*c)*d^3*e^4)*m^4 + (1260*(b^3 + 6*a*b*c)*e^7 + (
3*(b^2*c + a*c^2)*d*e^6 + (b^3 + 6*a*b*c)*e^7)*m^6 - 3*(5*b*c^2*d^2*e^5 - 19*(b^2*c + a*c^2)*d*e^6 - 8*(b^3 +
6*a*b*c)*e^7)*m^5 + (30*c^3*d^3*e^4 - 195*b*c^2*d^2*e^5 + 393*(b^2*c + a*c^2)*d*e^6 + 226*(b^3 + 6*a*b*c)*e^7)
*m^4 + 3*(60*c^3*d^3*e^4 - 265*b*c^2*d^2*e^5 + 401*(b^2*c + a*c^2)*d*e^6 + 352*(b^3 + 6*a*b*c)*e^7)*m^3 + 5*(6
6*c^3*d^3*e^4 - 249*b*c^2*d^2*e^5 + 324*(b^2*c + a*c^2)*d*e^6 + 509*(b^3 + 6*a*b*c)*e^7)*m^2 + 18*(10*c^3*d^3*
e^4 - 35*b*c^2*d^2*e^5 + 42*(b^2*c + a*c^2)*d*e^6 + 164*(b^3 + 6*a*b*c)*e^7)*m)*x^4 - 3*(245*a^2*b*d^2*e^5 - 5
55*a^3*d*e^6 + 2*(b^3 + 6*a*b*c)*d^4*e^3 - 44*(a*b^2 + a^2*c)*d^3*e^4)*m^3 + (5040*(a*b^2 + a^2*c)*e^7 + ((b^3
+ 6*a*b*c)*d*e^6 + 3*(a*b^2 + a^2*c)*e^7)*m^6 - 3*(4*(b^2*c + a*c^2)*d^2*e^5 - 7*(b^3 + 6*a*b*c)*d*e^6 - 25*(
a*b^2 + a^2*c)*e^7)*m^5 + (60*b*c^2*d^3*e^4 - 192*(b^2*c + a*c^2)*d^2*e^5 + 163*(b^3 + 6*a*b*c)*d*e^6 + 741*(a
*b^2 + a^2*c)*e^7)*m^4 - 3*(40*c^3*d^4*e^3 - 200*b*c^2*d^3*e^4 + 332*(b^2*c + a*c^2)*d^2*e^5 - 189*(b^3 + 6*a*
b*c)*d*e^6 - 1219*(a*b^2 + a^2*c)*e^7)*m^3 - 4*(90*c^3*d^4*e^3 - 345*b*c^2*d^3*e^4 + 456*(b^2*c + a*c^2)*d^2*e
^5 - 211*(b^3 + 6*a*b*c)*d*e^6 - 2334*(a*b^2 + a^2*c)*e^7)*m^2 - 12*(20*c^3*d^4*e^3 - 70*b*c^2*d^3*e^4 + 84*(b
^2*c + a*c^2)*d^2*e^5 - 35*(b^3 + 6*a*b*c)*d*e^6 - 949*(a*b^2 + a^2*c)*e^7)*m)*x^3 - (3525*a^2*b*d^2*e^5 - 510
4*a^3*d*e^6 - 72*(b^2*c + a*c^2)*d^5*e^2 + 108*(b^3 + 6*a*b*c)*d^4*e^3 - 1074*(a*b^2 + a^2*c)*d^3*e^4)*m^2 + 3
*(2520*a^2*b*e^7 + (a^2*b*e^7 + (a*b^2 + a^2*c)*d*e^6)*m^6 + (26*a^2*b*e^7 - (b^3 + 6*a*b*c)*d^2*e^5 + 23*(a*b
^2 + a^2*c)*d*e^6)*m^5 + (270*a^2*b*e^7 + 12*(b^2*c + a*c^2)*d^3*e^4 - 19*(b^3 + 6*a*b*c)*d^2*e^5 + 201*(a*b^2
+ a^2*c)*d*e^6)*m^4 - (60*b*c^2*d^4*e^3 - 1420*a^2*b*e^7 - 168*(b^2*c + a*c^2)*d^3*e^4 + 125*(b^3 + 6*a*b*c)*
d^2*e^5 - 817*(a*b^2 + a^2*c)*d*e^6)*m^3 + (120*c^3*d^5*e^2 - 480*b*c^2*d^4*e^3 + 3929*a^2*b*e^7 + 660*(b^2*c
+ a*c^2)*d^3*e^4 - 317*(b^3 + 6*a*b*c)*d^2*e^5 + 1478*(a*b^2 + a^2*c)*d*e^6)*m^2 + 6*(20*c^3*d^5*e^2 - 70*b*c^
2*d^4*e^3 + 879*a^2*b*e^7 + 84*(b^2*c + a*c^2)*d^3*e^4 - 35*(b^3 + 6*a*b*c)*d^2*e^5 + 140*(a*b^2 + a^2*c)*d*e^
6)*m)*x^2 - 6*(60*b*c^2*d^6*e + 1377*a^2*b*d^2*e^5 - 1338*a^3*d*e^6 - 156*(b^2*c + a*c^2)*d^5*e^2 + 107*(b^3 +
6*a*b*c)*d^4*e^3 - 638*(a*b^2 + a^2*c)*d^3*e^4)*m + (5040*a^3*e^7 + (3*a^2*b*d*e^6 + a^3*e^7)*m^6 + 3*(25*a^2
*b*d*e^6 + 9*a^3*e^7 - 2*(a*b^2 + a^2*c)*d^2*e^5)*m^5 + (735*a^2*b*d*e^6 + 295*a^3*e^7 + 6*(b^3 + 6*a*b*c)*d^3
*e^4 - 132*(a*b^2 + a^2*c)*d^2*e^5)*m^4 + 3*(1175*a^2*b*d*e^6 + 555*a^3*e^7 - 24*(b^2*c + a*c^2)*d^4*e^3 + 36*
(b^3 + 6*a*b*c)*d^3*e^4 - 358*(a*b^2 + a^2*c)*d^2*e^5)*m^3 + 2*(180*b*c^2*d^5*e^2 + 4131*a^2*b*d*e^6 + 2552*a^
3*e^7 - 468*(b^2*c + a*c^2)*d^4*e^3 + 321*(b^3 + 6*a*b*c)*d^3*e^4 - 1914*(a*b^2 + a^2*c)*d^2*e^5)*m^2 - 36*(20
*c^3*d^6*e - 70*b*c^2*d^5*e^2 - 210*a^2*b*d*e^6 - 223*a^3*e^7 + 84*(b^2*c + a*c^2)*d^4*e^3 - 35*(b^3 + 6*a*b*c
)*d^3*e^4 + 140*(a*b^2 + a^2*c)*d^2*e^5)*m)*x)*(e*x + d)^m/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4
+ 6769*e^7*m^3 + 13132*e^7*m^2 + 13068*e^7*m + 5040*e^7)

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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((e*x+d)**m*(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.51168, size = 7272, normalized size = 23.84 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

((x*e + d)^m*c^3*m^6*x^7*e^7 + (x*e + d)^m*c^3*d*m^6*x^6*e^6 + 3*(x*e + d)^m*b*c^2*m^6*x^6*e^7 + 21*(x*e + d)^
m*c^3*m^5*x^7*e^7 + 3*(x*e + d)^m*b*c^2*d*m^6*x^5*e^6 + 15*(x*e + d)^m*c^3*d*m^5*x^6*e^6 - 6*(x*e + d)^m*c^3*d
^2*m^5*x^5*e^5 + 3*(x*e + d)^m*b^2*c*m^6*x^5*e^7 + 3*(x*e + d)^m*a*c^2*m^6*x^5*e^7 + 66*(x*e + d)^m*b*c^2*m^5*
x^6*e^7 + 175*(x*e + d)^m*c^3*m^4*x^7*e^7 + 3*(x*e + d)^m*b^2*c*d*m^6*x^4*e^6 + 3*(x*e + d)^m*a*c^2*d*m^6*x^4*
e^6 + 51*(x*e + d)^m*b*c^2*d*m^5*x^5*e^6 + 85*(x*e + d)^m*c^3*d*m^4*x^6*e^6 - 15*(x*e + d)^m*b*c^2*d^2*m^5*x^4
*e^5 - 60*(x*e + d)^m*c^3*d^2*m^4*x^5*e^5 + 30*(x*e + d)^m*c^3*d^3*m^4*x^4*e^4 + (x*e + d)^m*b^3*m^6*x^4*e^7 +
6*(x*e + d)^m*a*b*c*m^6*x^4*e^7 + 69*(x*e + d)^m*b^2*c*m^5*x^5*e^7 + 69*(x*e + d)^m*a*c^2*m^5*x^5*e^7 + 570*(
x*e + d)^m*b*c^2*m^4*x^6*e^7 + 735*(x*e + d)^m*c^3*m^3*x^7*e^7 + (x*e + d)^m*b^3*d*m^6*x^3*e^6 + 6*(x*e + d)^m
*a*b*c*d*m^6*x^3*e^6 + 57*(x*e + d)^m*b^2*c*d*m^5*x^4*e^6 + 57*(x*e + d)^m*a*c^2*d*m^5*x^4*e^6 + 315*(x*e + d)
^m*b*c^2*d*m^4*x^5*e^6 + 225*(x*e + d)^m*c^3*d*m^3*x^6*e^6 - 12*(x*e + d)^m*b^2*c*d^2*m^5*x^3*e^5 - 12*(x*e +
d)^m*a*c^2*d^2*m^5*x^3*e^5 - 195*(x*e + d)^m*b*c^2*d^2*m^4*x^4*e^5 - 210*(x*e + d)^m*c^3*d^2*m^3*x^5*e^5 + 60*
(x*e + d)^m*b*c^2*d^3*m^4*x^3*e^4 + 180*(x*e + d)^m*c^3*d^3*m^3*x^4*e^4 - 120*(x*e + d)^m*c^3*d^4*m^3*x^3*e^3
+ 3*(x*e + d)^m*a*b^2*m^6*x^3*e^7 + 3*(x*e + d)^m*a^2*c*m^6*x^3*e^7 + 24*(x*e + d)^m*b^3*m^5*x^4*e^7 + 144*(x*
e + d)^m*a*b*c*m^5*x^4*e^7 + 621*(x*e + d)^m*b^2*c*m^4*x^5*e^7 + 621*(x*e + d)^m*a*c^2*m^4*x^5*e^7 + 2460*(x*e
+ d)^m*b*c^2*m^3*x^6*e^7 + 1624*(x*e + d)^m*c^3*m^2*x^7*e^7 + 3*(x*e + d)^m*a*b^2*d*m^6*x^2*e^6 + 3*(x*e + d)
^m*a^2*c*d*m^6*x^2*e^6 + 21*(x*e + d)^m*b^3*d*m^5*x^3*e^6 + 126*(x*e + d)^m*a*b*c*d*m^5*x^3*e^6 + 393*(x*e + d
)^m*b^2*c*d*m^4*x^4*e^6 + 393*(x*e + d)^m*a*c^2*d*m^4*x^4*e^6 + 885*(x*e + d)^m*b*c^2*d*m^3*x^5*e^6 + 274*(x*e
+ d)^m*c^3*d*m^2*x^6*e^6 - 3*(x*e + d)^m*b^3*d^2*m^5*x^2*e^5 - 18*(x*e + d)^m*a*b*c*d^2*m^5*x^2*e^5 - 192*(x*
e + d)^m*b^2*c*d^2*m^4*x^3*e^5 - 192*(x*e + d)^m*a*c^2*d^2*m^4*x^3*e^5 - 795*(x*e + d)^m*b*c^2*d^2*m^3*x^4*e^5
- 300*(x*e + d)^m*c^3*d^2*m^2*x^5*e^5 + 36*(x*e + d)^m*b^2*c*d^3*m^4*x^2*e^4 + 36*(x*e + d)^m*a*c^2*d^3*m^4*x
^2*e^4 + 600*(x*e + d)^m*b*c^2*d^3*m^3*x^3*e^4 + 330*(x*e + d)^m*c^3*d^3*m^2*x^4*e^4 - 180*(x*e + d)^m*b*c^2*d
^4*m^3*x^2*e^3 - 360*(x*e + d)^m*c^3*d^4*m^2*x^3*e^3 + 360*(x*e + d)^m*c^3*d^5*m^2*x^2*e^2 + 3*(x*e + d)^m*a^2
*b*m^6*x^2*e^7 + 75*(x*e + d)^m*a*b^2*m^5*x^3*e^7 + 75*(x*e + d)^m*a^2*c*m^5*x^3*e^7 + 226*(x*e + d)^m*b^3*m^4
*x^4*e^7 + 1356*(x*e + d)^m*a*b*c*m^4*x^4*e^7 + 2775*(x*e + d)^m*b^2*c*m^3*x^5*e^7 + 2775*(x*e + d)^m*a*c^2*m^
3*x^5*e^7 + 5547*(x*e + d)^m*b*c^2*m^2*x^6*e^7 + 1764*(x*e + d)^m*c^3*m*x^7*e^7 + 3*(x*e + d)^m*a^2*b*d*m^6*x*
e^6 + 69*(x*e + d)^m*a*b^2*d*m^5*x^2*e^6 + 69*(x*e + d)^m*a^2*c*d*m^5*x^2*e^6 + 163*(x*e + d)^m*b^3*d*m^4*x^3*
e^6 + 978*(x*e + d)^m*a*b*c*d*m^4*x^3*e^6 + 1203*(x*e + d)^m*b^2*c*d*m^3*x^4*e^6 + 1203*(x*e + d)^m*a*c^2*d*m^
3*x^4*e^6 + 1122*(x*e + d)^m*b*c^2*d*m^2*x^5*e^6 + 120*(x*e + d)^m*c^3*d*m*x^6*e^6 - 6*(x*e + d)^m*a*b^2*d^2*m
^5*x*e^5 - 6*(x*e + d)^m*a^2*c*d^2*m^5*x*e^5 - 57*(x*e + d)^m*b^3*d^2*m^4*x^2*e^5 - 342*(x*e + d)^m*a*b*c*d^2*
m^4*x^2*e^5 - 996*(x*e + d)^m*b^2*c*d^2*m^3*x^3*e^5 - 996*(x*e + d)^m*a*c^2*d^2*m^3*x^3*e^5 - 1245*(x*e + d)^m
*b*c^2*d^2*m^2*x^4*e^5 - 144*(x*e + d)^m*c^3*d^2*m*x^5*e^5 + 6*(x*e + d)^m*b^3*d^3*m^4*x*e^4 + 36*(x*e + d)^m*
a*b*c*d^3*m^4*x*e^4 + 504*(x*e + d)^m*b^2*c*d^3*m^3*x^2*e^4 + 504*(x*e + d)^m*a*c^2*d^3*m^3*x^2*e^4 + 1380*(x*
e + d)^m*b*c^2*d^3*m^2*x^3*e^4 + 180*(x*e + d)^m*c^3*d^3*m*x^4*e^4 - 72*(x*e + d)^m*b^2*c*d^4*m^3*x*e^3 - 72*(
x*e + d)^m*a*c^2*d^4*m^3*x*e^3 - 1440*(x*e + d)^m*b*c^2*d^4*m^2*x^2*e^3 - 240*(x*e + d)^m*c^3*d^4*m*x^3*e^3 +
360*(x*e + d)^m*b*c^2*d^5*m^2*x*e^2 + 360*(x*e + d)^m*c^3*d^5*m*x^2*e^2 - 720*(x*e + d)^m*c^3*d^6*m*x*e + (x*e
+ d)^m*a^3*m^6*x*e^7 + 78*(x*e + d)^m*a^2*b*m^5*x^2*e^7 + 741*(x*e + d)^m*a*b^2*m^4*x^3*e^7 + 741*(x*e + d)^m
*a^2*c*m^4*x^3*e^7 + 1056*(x*e + d)^m*b^3*m^3*x^4*e^7 + 6336*(x*e + d)^m*a*b*c*m^3*x^4*e^7 + 6432*(x*e + d)^m*
b^2*c*m^2*x^5*e^7 + 6432*(x*e + d)^m*a*c^2*m^2*x^5*e^7 + 6114*(x*e + d)^m*b*c^2*m*x^6*e^7 + 720*(x*e + d)^m*c^
3*x^7*e^7 + (x*e + d)^m*a^3*d*m^6*e^6 + 75*(x*e + d)^m*a^2*b*d*m^5*x*e^6 + 603*(x*e + d)^m*a*b^2*d*m^4*x^2*e^6
+ 603*(x*e + d)^m*a^2*c*d*m^4*x^2*e^6 + 567*(x*e + d)^m*b^3*d*m^3*x^3*e^6 + 3402*(x*e + d)^m*a*b*c*d*m^3*x^3*
e^6 + 1620*(x*e + d)^m*b^2*c*d*m^2*x^4*e^6 + 1620*(x*e + d)^m*a*c^2*d*m^2*x^4*e^6 + 504*(x*e + d)^m*b*c^2*d*m*
x^5*e^6 - 3*(x*e + d)^m*a^2*b*d^2*m^5*e^5 - 132*(x*e + d)^m*a*b^2*d^2*m^4*x*e^5 - 132*(x*e + d)^m*a^2*c*d^2*m^
4*x*e^5 - 375*(x*e + d)^m*b^3*d^2*m^3*x^2*e^5 - 2250*(x*e + d)^m*a*b*c*d^2*m^3*x^2*e^5 - 1824*(x*e + d)^m*b^2*
c*d^2*m^2*x^3*e^5 - 1824*(x*e + d)^m*a*c^2*d^2*m^2*x^3*e^5 - 630*(x*e + d)^m*b*c^2*d^2*m*x^4*e^5 + 6*(x*e + d)
^m*a*b^2*d^3*m^4*e^4 + 6*(x*e + d)^m*a^2*c*d^3*m^4*e^4 + 108*(x*e + d)^m*b^3*d^3*m^3*x*e^4 + 648*(x*e + d)^m*a
*b*c*d^3*m^3*x*e^4 + 1980*(x*e + d)^m*b^2*c*d^3*m^2*x^2*e^4 + 1980*(x*e + d)^m*a*c^2*d^3*m^2*x^2*e^4 + 840*(x*
e + d)^m*b*c^2*d^3*m*x^3*e^4 - 6*(x*e + d)^m*b^3*d^4*m^3*e^3 - 36*(x*e + d)^m*a*b*c*d^4*m^3*e^3 - 936*(x*e + d
)^m*b^2*c*d^4*m^2*x*e^3 - 936*(x*e + d)^m*a*c^2*d^4*m^2*x*e^3 - 1260*(x*e + d)^m*b*c^2*d^4*m*x^2*e^3 + 72*(x*e
+ d)^m*b^2*c*d^5*m^2*e^2 + 72*(x*e + d)^m*a*c^2*d^5*m^2*e^2 + 2520*(x*e + d)^m*b*c^2*d^5*m*x*e^2 - 360*(x*e +
d)^m*b*c^2*d^6*m*e + 720*(x*e + d)^m*c^3*d^7 + 27*(x*e + d)^m*a^3*m^5*x*e^7 + 810*(x*e + d)^m*a^2*b*m^4*x^2*e
^7 + 3657*(x*e + d)^m*a*b^2*m^3*x^3*e^7 + 3657*(x*e + d)^m*a^2*c*m^3*x^3*e^7 + 2545*(x*e + d)^m*b^3*m^2*x^4*e^
7 + 15270*(x*e + d)^m*a*b*c*m^2*x^4*e^7 + 7236*(x*e + d)^m*b^2*c*m*x^5*e^7 + 7236*(x*e + d)^m*a*c^2*m*x^5*e^7
+ 2520*(x*e + d)^m*b*c^2*x^6*e^7 + 27*(x*e + d)^m*a^3*d*m^5*e^6 + 735*(x*e + d)^m*a^2*b*d*m^4*x*e^6 + 2451*(x*
e + d)^m*a*b^2*d*m^3*x^2*e^6 + 2451*(x*e + d)^m*a^2*c*d*m^3*x^2*e^6 + 844*(x*e + d)^m*b^3*d*m^2*x^3*e^6 + 5064
*(x*e + d)^m*a*b*c*d*m^2*x^3*e^6 + 756*(x*e + d)^m*b^2*c*d*m*x^4*e^6 + 756*(x*e + d)^m*a*c^2*d*m*x^4*e^6 - 75*
(x*e + d)^m*a^2*b*d^2*m^4*e^5 - 1074*(x*e + d)^m*a*b^2*d^2*m^3*x*e^5 - 1074*(x*e + d)^m*a^2*c*d^2*m^3*x*e^5 -
951*(x*e + d)^m*b^3*d^2*m^2*x^2*e^5 - 5706*(x*e + d)^m*a*b*c*d^2*m^2*x^2*e^5 - 1008*(x*e + d)^m*b^2*c*d^2*m*x^
3*e^5 - 1008*(x*e + d)^m*a*c^2*d^2*m*x^3*e^5 + 132*(x*e + d)^m*a*b^2*d^3*m^3*e^4 + 132*(x*e + d)^m*a^2*c*d^3*m
^3*e^4 + 642*(x*e + d)^m*b^3*d^3*m^2*x*e^4 + 3852*(x*e + d)^m*a*b*c*d^3*m^2*x*e^4 + 1512*(x*e + d)^m*b^2*c*d^3
*m*x^2*e^4 + 1512*(x*e + d)^m*a*c^2*d^3*m*x^2*e^4 - 108*(x*e + d)^m*b^3*d^4*m^2*e^3 - 648*(x*e + d)^m*a*b*c*d^
4*m^2*e^3 - 3024*(x*e + d)^m*b^2*c*d^4*m*x*e^3 - 3024*(x*e + d)^m*a*c^2*d^4*m*x*e^3 + 936*(x*e + d)^m*b^2*c*d^
5*m*e^2 + 936*(x*e + d)^m*a*c^2*d^5*m*e^2 - 2520*(x*e + d)^m*b*c^2*d^6*e + 295*(x*e + d)^m*a^3*m^4*x*e^7 + 426
0*(x*e + d)^m*a^2*b*m^3*x^2*e^7 + 9336*(x*e + d)^m*a*b^2*m^2*x^3*e^7 + 9336*(x*e + d)^m*a^2*c*m^2*x^3*e^7 + 29
52*(x*e + d)^m*b^3*m*x^4*e^7 + 17712*(x*e + d)^m*a*b*c*m*x^4*e^7 + 3024*(x*e + d)^m*b^2*c*x^5*e^7 + 3024*(x*e
+ d)^m*a*c^2*x^5*e^7 + 295*(x*e + d)^m*a^3*d*m^4*e^6 + 3525*(x*e + d)^m*a^2*b*d*m^3*x*e^6 + 4434*(x*e + d)^m*a
*b^2*d*m^2*x^2*e^6 + 4434*(x*e + d)^m*a^2*c*d*m^2*x^2*e^6 + 420*(x*e + d)^m*b^3*d*m*x^3*e^6 + 2520*(x*e + d)^m
*a*b*c*d*m*x^3*e^6 - 735*(x*e + d)^m*a^2*b*d^2*m^3*e^5 - 3828*(x*e + d)^m*a*b^2*d^2*m^2*x*e^5 - 3828*(x*e + d)
^m*a^2*c*d^2*m^2*x*e^5 - 630*(x*e + d)^m*b^3*d^2*m*x^2*e^5 - 3780*(x*e + d)^m*a*b*c*d^2*m*x^2*e^5 + 1074*(x*e
+ d)^m*a*b^2*d^3*m^2*e^4 + 1074*(x*e + d)^m*a^2*c*d^3*m^2*e^4 + 1260*(x*e + d)^m*b^3*d^3*m*x*e^4 + 7560*(x*e +
d)^m*a*b*c*d^3*m*x*e^4 - 642*(x*e + d)^m*b^3*d^4*m*e^3 - 3852*(x*e + d)^m*a*b*c*d^4*m*e^3 + 3024*(x*e + d)^m*
b^2*c*d^5*e^2 + 3024*(x*e + d)^m*a*c^2*d^5*e^2 + 1665*(x*e + d)^m*a^3*m^3*x*e^7 + 11787*(x*e + d)^m*a^2*b*m^2*
x^2*e^7 + 11388*(x*e + d)^m*a*b^2*m*x^3*e^7 + 11388*(x*e + d)^m*a^2*c*m*x^3*e^7 + 1260*(x*e + d)^m*b^3*x^4*e^7
+ 7560*(x*e + d)^m*a*b*c*x^4*e^7 + 1665*(x*e + d)^m*a^3*d*m^3*e^6 + 8262*(x*e + d)^m*a^2*b*d*m^2*x*e^6 + 2520
*(x*e + d)^m*a*b^2*d*m*x^2*e^6 + 2520*(x*e + d)^m*a^2*c*d*m*x^2*e^6 - 3525*(x*e + d)^m*a^2*b*d^2*m^2*e^5 - 504
0*(x*e + d)^m*a*b^2*d^2*m*x*e^5 - 5040*(x*e + d)^m*a^2*c*d^2*m*x*e^5 + 3828*(x*e + d)^m*a*b^2*d^3*m*e^4 + 3828
*(x*e + d)^m*a^2*c*d^3*m*e^4 - 1260*(x*e + d)^m*b^3*d^4*e^3 - 7560*(x*e + d)^m*a*b*c*d^4*e^3 + 5104*(x*e + d)^
m*a^3*m^2*x*e^7 + 15822*(x*e + d)^m*a^2*b*m*x^2*e^7 + 5040*(x*e + d)^m*a*b^2*x^3*e^7 + 5040*(x*e + d)^m*a^2*c*
x^3*e^7 + 5104*(x*e + d)^m*a^3*d*m^2*e^6 + 7560*(x*e + d)^m*a^2*b*d*m*x*e^6 - 8262*(x*e + d)^m*a^2*b*d^2*m*e^5
+ 5040*(x*e + d)^m*a*b^2*d^3*e^4 + 5040*(x*e + d)^m*a^2*c*d^3*e^4 + 8028*(x*e + d)^m*a^3*m*x*e^7 + 7560*(x*e
+ d)^m*a^2*b*x^2*e^7 + 8028*(x*e + d)^m*a^3*d*m*e^6 - 7560*(x*e + d)^m*a^2*b*d^2*e^5 + 5040*(x*e + d)^m*a^3*x*
e^7 + 5040*(x*e + d)^m*a^3*d*e^6)/(m^7*e^7 + 28*m^6*e^7 + 322*m^5*e^7 + 1960*m^4*e^7 + 6769*m^3*e^7 + 13132*m^
2*e^7 + 13068*m*e^7 + 5040*e^7)