### 3.37 $$\int (d x)^m (A+B x+C x^2) (a+b x^2+c x^4)^3 \, dx$$

Optimal. Leaf size=399 $\frac{a^2 (d x)^{m+3} (a C+3 A b)}{d^3 (m+3)}+\frac{a^3 A (d x)^{m+1}}{d (m+1)}+\frac{3 a^2 b B (d x)^{m+4}}{d^4 (m+4)}+\frac{a^3 B (d x)^{m+2}}{d^2 (m+2)}+\frac{3 a (d x)^{m+5} \left (A \left (a c+b^2\right )+a b C\right )}{d^5 (m+5)}+\frac{(d x)^{m+7} \left (A \left (6 a b c+b^3\right )+3 a C \left (a c+b^2\right )\right )}{d^7 (m+7)}+\frac{(d x)^{m+9} \left (3 A c \left (a c+b^2\right )+b C \left (6 a c+b^2\right )\right )}{d^9 (m+9)}+\frac{3 c (d x)^{m+11} \left (C \left (a c+b^2\right )+A b c\right )}{d^{11} (m+11)}+\frac{3 a B \left (a c+b^2\right ) (d x)^{m+6}}{d^6 (m+6)}+\frac{b B \left (6 a c+b^2\right ) (d x)^{m+8}}{d^8 (m+8)}+\frac{3 B c \left (a c+b^2\right ) (d x)^{m+10}}{d^{10} (m+10)}+\frac{c^2 (d x)^{m+13} (A c+3 b C)}{d^{13} (m+13)}+\frac{3 b B c^2 (d x)^{m+12}}{d^{12} (m+12)}+\frac{B c^3 (d x)^{m+14}}{d^{14} (m+14)}+\frac{c^3 C (d x)^{m+15}}{d^{15} (m+15)}$

[Out]

(a^3*A*(d*x)^(1 + m))/(d*(1 + m)) + (a^3*B*(d*x)^(2 + m))/(d^2*(2 + m)) + (a^2*(3*A*b + a*C)*(d*x)^(3 + m))/(d
^3*(3 + m)) + (3*a^2*b*B*(d*x)^(4 + m))/(d^4*(4 + m)) + (3*a*(A*(b^2 + a*c) + a*b*C)*(d*x)^(5 + m))/(d^5*(5 +
m)) + (3*a*B*(b^2 + a*c)*(d*x)^(6 + m))/(d^6*(6 + m)) + ((A*(b^3 + 6*a*b*c) + 3*a*(b^2 + a*c)*C)*(d*x)^(7 + m)
)/(d^7*(7 + m)) + (b*B*(b^2 + 6*a*c)*(d*x)^(8 + m))/(d^8*(8 + m)) + ((3*A*c*(b^2 + a*c) + b*(b^2 + 6*a*c)*C)*(
d*x)^(9 + m))/(d^9*(9 + m)) + (3*B*c*(b^2 + a*c)*(d*x)^(10 + m))/(d^10*(10 + m)) + (3*c*(A*b*c + (b^2 + a*c)*C
)*(d*x)^(11 + m))/(d^11*(11 + m)) + (3*b*B*c^2*(d*x)^(12 + m))/(d^12*(12 + m)) + (c^2*(A*c + 3*b*C)*(d*x)^(13
+ m))/(d^13*(13 + m)) + (B*c^3*(d*x)^(14 + m))/(d^14*(14 + m)) + (c^3*C*(d*x)^(15 + m))/(d^15*(15 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.424928, antiderivative size = 399, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.033, Rules used = {1628} $\frac{a^2 (d x)^{m+3} (a C+3 A b)}{d^3 (m+3)}+\frac{a^3 A (d x)^{m+1}}{d (m+1)}+\frac{3 a^2 b B (d x)^{m+4}}{d^4 (m+4)}+\frac{a^3 B (d x)^{m+2}}{d^2 (m+2)}+\frac{3 a (d x)^{m+5} \left (A \left (a c+b^2\right )+a b C\right )}{d^5 (m+5)}+\frac{(d x)^{m+7} \left (A \left (6 a b c+b^3\right )+3 a C \left (a c+b^2\right )\right )}{d^7 (m+7)}+\frac{(d x)^{m+9} \left (3 A c \left (a c+b^2\right )+b C \left (6 a c+b^2\right )\right )}{d^9 (m+9)}+\frac{3 c (d x)^{m+11} \left (C \left (a c+b^2\right )+A b c\right )}{d^{11} (m+11)}+\frac{3 a B \left (a c+b^2\right ) (d x)^{m+6}}{d^6 (m+6)}+\frac{b B \left (6 a c+b^2\right ) (d x)^{m+8}}{d^8 (m+8)}+\frac{3 B c \left (a c+b^2\right ) (d x)^{m+10}}{d^{10} (m+10)}+\frac{c^2 (d x)^{m+13} (A c+3 b C)}{d^{13} (m+13)}+\frac{3 b B c^2 (d x)^{m+12}}{d^{12} (m+12)}+\frac{B c^3 (d x)^{m+14}}{d^{14} (m+14)}+\frac{c^3 C (d x)^{m+15}}{d^{15} (m+15)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d*x)^m*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^3,x]

[Out]

(a^3*A*(d*x)^(1 + m))/(d*(1 + m)) + (a^3*B*(d*x)^(2 + m))/(d^2*(2 + m)) + (a^2*(3*A*b + a*C)*(d*x)^(3 + m))/(d
^3*(3 + m)) + (3*a^2*b*B*(d*x)^(4 + m))/(d^4*(4 + m)) + (3*a*(A*(b^2 + a*c) + a*b*C)*(d*x)^(5 + m))/(d^5*(5 +
m)) + (3*a*B*(b^2 + a*c)*(d*x)^(6 + m))/(d^6*(6 + m)) + ((A*(b^3 + 6*a*b*c) + 3*a*(b^2 + a*c)*C)*(d*x)^(7 + m)
)/(d^7*(7 + m)) + (b*B*(b^2 + 6*a*c)*(d*x)^(8 + m))/(d^8*(8 + m)) + ((3*A*c*(b^2 + a*c) + b*(b^2 + 6*a*c)*C)*(
d*x)^(9 + m))/(d^9*(9 + m)) + (3*B*c*(b^2 + a*c)*(d*x)^(10 + m))/(d^10*(10 + m)) + (3*c*(A*b*c + (b^2 + a*c)*C
)*(d*x)^(11 + m))/(d^11*(11 + m)) + (3*b*B*c^2*(d*x)^(12 + m))/(d^12*(12 + m)) + (c^2*(A*c + 3*b*C)*(d*x)^(13
+ m))/(d^13*(13 + m)) + (B*c^3*(d*x)^(14 + m))/(d^14*(14 + m)) + (c^3*C*(d*x)^(15 + m))/(d^15*(15 + m))

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 A (d x)^m+\frac{a^3 B (d x)^{1+m}}{d}+\frac{a^2 (3 A b+a C) (d x)^{2+m}}{d^2}+\frac{3 a^2 b B (d x)^{3+m}}{d^3}+\frac{3 a \left (A \left (b^2+a c\right )+a b C\right ) (d x)^{4+m}}{d^4}+\frac{3 a B \left (b^2+a c\right ) (d x)^{5+m}}{d^5}+\frac{\left (A \left (b^3+6 a b c\right )+3 a \left (b^2+a c\right ) C\right ) (d x)^{6+m}}{d^6}+\frac{b B \left (b^2+6 a c\right ) (d x)^{7+m}}{d^7}+\frac{\left (3 A c \left (b^2+a c\right )+b \left (b^2+6 a c\right ) C\right ) (d x)^{8+m}}{d^8}+\frac{3 B c \left (b^2+a c\right ) (d x)^{9+m}}{d^9}+\frac{3 c \left (A b c+\left (b^2+a c\right ) C\right ) (d x)^{10+m}}{d^{10}}+\frac{3 b B c^2 (d x)^{11+m}}{d^{11}}+\frac{c^2 (A c+3 b C) (d x)^{12+m}}{d^{12}}+\frac{B c^3 (d x)^{13+m}}{d^{13}}+\frac{c^3 C (d x)^{14+m}}{d^{14}}\right ) \, dx\\ &=\frac{a^3 A (d x)^{1+m}}{d (1+m)}+\frac{a^3 B (d x)^{2+m}}{d^2 (2+m)}+\frac{a^2 (3 A b+a C) (d x)^{3+m}}{d^3 (3+m)}+\frac{3 a^2 b B (d x)^{4+m}}{d^4 (4+m)}+\frac{3 a \left (A \left (b^2+a c\right )+a b C\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac{3 a B \left (b^2+a c\right ) (d x)^{6+m}}{d^6 (6+m)}+\frac{\left (A \left (b^3+6 a b c\right )+3 a \left (b^2+a c\right ) C\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac{b B \left (b^2+6 a c\right ) (d x)^{8+m}}{d^8 (8+m)}+\frac{\left (3 A c \left (b^2+a c\right )+b \left (b^2+6 a c\right ) C\right ) (d x)^{9+m}}{d^9 (9+m)}+\frac{3 B c \left (b^2+a c\right ) (d x)^{10+m}}{d^{10} (10+m)}+\frac{3 c \left (A b c+\left (b^2+a c\right ) C\right ) (d x)^{11+m}}{d^{11} (11+m)}+\frac{3 b B c^2 (d x)^{12+m}}{d^{12} (12+m)}+\frac{c^2 (A c+3 b C) (d x)^{13+m}}{d^{13} (13+m)}+\frac{B c^3 (d x)^{14+m}}{d^{14} (14+m)}+\frac{c^3 C (d x)^{15+m}}{d^{15} (15+m)}\\ \end{align*}

Mathematica [A]  time = 1.27055, size = 296, normalized size = 0.74 $x (d x)^m \left (\frac{a^2 x^2 (a C+3 A b)}{m+3}+\frac{a^3 A}{m+1}+\frac{3 a^2 b B x^3}{m+4}+\frac{a^3 B x}{m+2}+\frac{3 c x^{10} \left (C \left (a c+b^2\right )+A b c\right )}{m+11}+\frac{x^8 \left (3 A c \left (a c+b^2\right )+b C \left (6 a c+b^2\right )\right )}{m+9}+\frac{x^6 \left (A \left (6 a b c+b^3\right )+3 a C \left (a c+b^2\right )\right )}{m+7}+\frac{3 a x^4 \left (A \left (a c+b^2\right )+a b C\right )}{m+5}+\frac{3 B c x^9 \left (a c+b^2\right )}{m+10}+\frac{b B x^7 \left (6 a c+b^2\right )}{m+8}+\frac{3 a B x^5 \left (a c+b^2\right )}{m+6}+\frac{c^2 x^{12} (A c+3 b C)}{m+13}+\frac{3 b B c^2 x^{11}}{m+12}+\frac{B c^3 x^{13}}{m+14}+\frac{c^3 C x^{14}}{m+15}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d*x)^m*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^3,x]

[Out]

x*(d*x)^m*((a^3*A)/(1 + m) + (a^3*B*x)/(2 + m) + (a^2*(3*A*b + a*C)*x^2)/(3 + m) + (3*a^2*b*B*x^3)/(4 + m) + (
3*a*(A*(b^2 + a*c) + a*b*C)*x^4)/(5 + m) + (3*a*B*(b^2 + a*c)*x^5)/(6 + m) + ((A*(b^3 + 6*a*b*c) + 3*a*(b^2 +
a*c)*C)*x^6)/(7 + m) + (b*B*(b^2 + 6*a*c)*x^7)/(8 + m) + ((3*A*c*(b^2 + a*c) + b*(b^2 + 6*a*c)*C)*x^8)/(9 + m)
+ (3*B*c*(b^2 + a*c)*x^9)/(10 + m) + (3*c*(A*b*c + (b^2 + a*c)*C)*x^10)/(11 + m) + (3*b*B*c^2*x^11)/(12 + m)
+ (c^2*(A*c + 3*b*C)*x^12)/(13 + m) + (B*c^3*x^13)/(14 + m) + (c^3*C*x^14)/(15 + m))

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 5520, normalized size = 13.8 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(C*x^2+B*x+A)*(c*x^4+b*x^2+a)^3,x)

[Out]

result too large to display

________________________________________________________________________________________

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((d*x)^m*(C*x^2+B*x+A)*(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.67241, size = 11187, normalized size = 28.04 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(C*x^2+B*x+A)*(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

((C*c^3*m^14 + 105*C*c^3*m^13 + 5005*C*c^3*m^12 + 143325*C*c^3*m^11 + 2749747*C*c^3*m^10 + 37312275*C*c^3*m^9
+ 368411615*C*c^3*m^8 + 2681453775*C*c^3*m^7 + 14409322928*C*c^3*m^6 + 56663366760*C*c^3*m^5 + 159721605680*C*
c^3*m^4 + 310989260400*C*c^3*m^3 + 392156797824*C*c^3*m^2 + 283465647360*C*c^3*m + 87178291200*C*c^3)*x^15 + (
B*c^3*m^14 + 106*B*c^3*m^13 + 5096*B*c^3*m^12 + 147056*B*c^3*m^11 + 2840838*B*c^3*m^10 + 38786748*B*c^3*m^9 +
385081268*B*c^3*m^8 + 2816490248*B*c^3*m^7 + 15200266081*B*c^3*m^6 + 59999485546*B*c^3*m^5 + 169679309436*B*c^
3*m^4 + 331303013496*B*c^3*m^3 + 418753514880*B*c^3*m^2 + 303268406400*B*c^3*m + 93405312000*B*c^3)*x^14 + ((3
*C*b*c^2 + A*c^3)*m^14 + 107*(3*C*b*c^2 + A*c^3)*m^13 + 5189*(3*C*b*c^2 + A*c^3)*m^12 + 150943*(3*C*b*c^2 + A*
c^3)*m^11 + 2937363*(3*C*b*c^2 + A*c^3)*m^10 + 40372761*(3*C*b*c^2 + A*c^3)*m^9 + 403249847*(3*C*b*c^2 + A*c^3
)*m^8 + 2965379989*(3*C*b*c^2 + A*c^3)*m^7 + 16081189696*(3*C*b*c^2 + A*c^3)*m^6 + 63747744632*(3*C*b*c^2 + A*
c^3)*m^5 + 180951426864*(3*C*b*c^2 + A*c^3)*m^4 + 301771008000*C*b*c^2 + 100590336000*A*c^3 + 354444796368*(3*
C*b*c^2 + A*c^3)*m^3 + 449213351040*(3*C*b*c^2 + A*c^3)*m^2 + 326044051200*(3*C*b*c^2 + A*c^3)*m)*x^13 + 3*(B*
b*c^2*m^14 + 108*B*b*c^2*m^13 + 5284*B*b*c^2*m^12 + 154992*B*b*c^2*m^11 + 3039718*B*b*c^2*m^10 + 42081864*B*b*
c^2*m^9 + 423113372*B*b*c^2*m^8 + 3130267536*B*b*c^2*m^7 + 17067919121*B*b*c^2*m^6 + 67988181228*B*b*c^2*m^5 +
193813932344*B*b*c^2*m^4 + 381046157472*B*b*c^2*m^3 + 484441814160*B*b*c^2*m^2 + 352515844800*B*b*c^2*m + 108
972864000*B*b*c^2)*x^12 + 3*((C*b^2*c + (C*a + A*b)*c^2)*m^14 + 109*(C*b^2*c + (C*a + A*b)*c^2)*m^13 + 5381*(C
*b^2*c + (C*a + A*b)*c^2)*m^12 + 159209*(C*b^2*c + (C*a + A*b)*c^2)*m^11 + 3148323*(C*b^2*c + (C*a + A*b)*c^2)
*m^10 + 43926927*(C*b^2*c + (C*a + A*b)*c^2)*m^9 + 444899543*(C*b^2*c + (C*a + A*b)*c^2)*m^8 + 3313733027*(C*b
^2*c + (C*a + A*b)*c^2)*m^7 + 18180066256*(C*b^2*c + (C*a + A*b)*c^2)*m^6 + 72822481864*(C*b^2*c + (C*a + A*b)
*c^2)*m^5 + 208624806576*(C*b^2*c + (C*a + A*b)*c^2)*m^4 + 118879488000*C*b^2*c + 411940473264*(C*b^2*c + (C*a
+ A*b)*c^2)*m^3 + 118879488000*(C*a + A*b)*c^2 + 525650497920*(C*b^2*c + (C*a + A*b)*c^2)*m^2 + 383662137600*
(C*b^2*c + (C*a + A*b)*c^2)*m)*x^11 + 3*((B*b^2*c + B*a*c^2)*m^14 + 110*(B*b^2*c + B*a*c^2)*m^13 + 5480*(B*b^2
*c + B*a*c^2)*m^12 + 163600*(B*b^2*c + B*a*c^2)*m^11 + 3263622*(B*b^2*c + B*a*c^2)*m^10 + 45922260*(B*b^2*c +
B*a*c^2)*m^9 + 468873140*(B*b^2*c + B*a*c^2)*m^8 + 3518896600*(B*b^2*c + B*a*c^2)*m^7 + 19442163553*(B*b^2*c +
B*a*c^2)*m^6 + 78381575150*(B*b^2*c + B*a*c^2)*m^5 + 225856355580*(B*b^2*c + B*a*c^2)*m^4 + 130767436800*B*b^
2*c + 130767436800*B*a*c^2 + 448249789800*(B*b^2*c + B*a*c^2)*m^3 + 574497805824*(B*b^2*c + B*a*c^2)*m^2 + 420
839556480*(B*b^2*c + B*a*c^2)*m)*x^10 + ((C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^14 + 111*(C*b^3 + 3*A*a
*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^13 + 5581*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^12 + 168171*(C*b^3 + 3
*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^11 + 3386083*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^10 + 48083733*(
C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^9 + 495342143*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^8 + 3
749548713*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^7 + 20885191136*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^
2)*c)*m^6 + 84836490456*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^5 + 246143692976*(C*b^3 + 3*A*a*c^2 + 3*
(2*C*a*b + A*b^2)*c)*m^4 + 145297152000*C*b^3 + 435891456000*A*a*c^2 + 491520108816*(C*b^3 + 3*A*a*c^2 + 3*(2*
C*a*b + A*b^2)*c)*m^3 + 633314724480*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^2 + 435891456000*(2*C*a*b +
A*b^2)*c + 465985094400*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m)*x^9 + ((B*b^3 + 6*B*a*b*c)*m^14 + 112*
(B*b^3 + 6*B*a*b*c)*m^13 + 5684*(B*b^3 + 6*B*a*b*c)*m^12 + 172928*(B*b^3 + 6*B*a*b*c)*m^11 + 3516198*(B*b^3 +
6*B*a*b*c)*m^10 + 50428896*(B*b^3 + 6*B*a*b*c)*m^9 + 524664572*(B*b^3 + 6*B*a*b*c)*m^8 + 4010311424*(B*b^3 + 6
*B*a*b*c)*m^7 + 22548638161*(B*b^3 + 6*B*a*b*c)*m^6 + 92414105392*(B*b^3 + 6*B*a*b*c)*m^5 + 270359263944*(B*b^
3 + 6*B*a*b*c)*m^4 + 163459296000*B*b^3 + 980755776000*B*a*b*c + 543939234048*(B*b^3 + 6*B*a*b*c)*m^3 + 705481
831440*(B*b^3 + 6*B*a*b*c)*m^2 + 521962963200*(B*b^3 + 6*B*a*b*c)*m)*x^8 + ((3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*
A*a*b)*c)*m^14 + 113*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^13 + 5789*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2
*A*a*b)*c)*m^12 + 177877*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^11 + 3654483*(3*C*a*b^2 + A*b^3 + 3*(C*
a^2 + 2*A*a*b)*c)*m^10 + 52977099*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^9 + 557256047*(3*C*a*b^2 + A*b
^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^8 + 4306835671*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^7 + 24483279856*(3*
C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^6 + 101420251688*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^5 +
299730345264*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^4 + 560431872000*C*a*b^2 + 186810624000*A*b^3 + 608
700928752*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^3 + 796089202560*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a
*b)*c)*m^2 + 560431872000*(C*a^2 + 2*A*a*b)*c + 593193196800*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m)*x^
7 + 3*((B*a*b^2 + B*a^2*c)*m^14 + 114*(B*a*b^2 + B*a^2*c)*m^13 + 5896*(B*a*b^2 + B*a^2*c)*m^12 + 183024*(B*a*b
^2 + B*a^2*c)*m^11 + 3801478*(B*a*b^2 + B*a^2*c)*m^10 + 55749612*(B*a*b^2 + B*a^2*c)*m^9 + 593598068*(B*a*b^2
+ B*a^2*c)*m^8 + 4646039592*(B*a*b^2 + B*a^2*c)*m^7 + 26754892001*(B*a*b^2 + B*a^2*c)*m^6 + 112273858674*(B*a*
b^2 + B*a^2*c)*m^5 + 336028955036*(B*a*b^2 + B*a^2*c)*m^4 + 217945728000*B*a*b^2 + 217945728000*B*a^2*c + 6906
39615384*(B*a*b^2 + B*a^2*c)*m^3 + 913158011520*(B*a*b^2 + B*a^2*c)*m^2 + 686869545600*(B*a*b^2 + B*a^2*c)*m)*
x^6 + 3*((C*a^2*b + A*a*b^2 + A*a^2*c)*m^14 + 115*(C*a^2*b + A*a*b^2 + A*a^2*c)*m^13 + 6005*(C*a^2*b + A*a*b^2
+ A*a^2*c)*m^12 + 188375*(C*a^2*b + A*a*b^2 + A*a^2*c)*m^11 + 3957747*(C*a^2*b + A*a*b^2 + A*a^2*c)*m^10 + 58
769745*(C*a^2*b + A*a*b^2 + A*a^2*c)*m^9 + 634247015*(C*a^2*b + A*a*b^2 + A*a^2*c)*m^8 + 5036392925*(C*a^2*b +
A*a*b^2 + A*a^2*c)*m^7 + 29449164928*(C*a^2*b + A*a*b^2 + A*a^2*c)*m^6 + 125557386040*(C*a^2*b + A*a*b^2 + A*
a^2*c)*m^5 + 381885176880*(C*a^2*b + A*a*b^2 + A*a^2*c)*m^4 + 261534873600*C*a^2*b + 261534873600*A*a*b^2 + 26
1534873600*A*a^2*c + 797387461200*(C*a^2*b + A*a*b^2 + A*a^2*c)*m^3 + 1070058397824*(C*a^2*b + A*a*b^2 + A*a^2
*c)*m^2 + 815525625600*(C*a^2*b + A*a*b^2 + A*a^2*c)*m)*x^5 + 3*(B*a^2*b*m^14 + 116*B*a^2*b*m^13 + 6116*B*a^2*
b*m^12 + 193936*B*a^2*b*m^11 + 4123878*B*a^2*b*m^10 + 62062968*B*a^2*b*m^9 + 679843868*B*a^2*b*m^8 + 548825252
8*B*a^2*b*m^7 + 32678119441*B*a^2*b*m^6 + 142090732916*B*a^2*b*m^5 + 441309175416*B*a^2*b*m^4 + 941576643936*B
*a^2*b*m^3 + 1290689128080*B*a^2*b*m^2 + 1003061102400*B*a^2*b*m + 326918592000*B*a^2*b)*x^4 + ((C*a^3 + 3*A*a
^2*b)*m^14 + 117*(C*a^3 + 3*A*a^2*b)*m^13 + 6229*(C*a^3 + 3*A*a^2*b)*m^12 + 199713*(C*a^3 + 3*A*a^2*b)*m^11 +
4300483*(C*a^3 + 3*A*a^2*b)*m^10 + 65657031*(C*a^3 + 3*A*a^2*b)*m^9 + 731124647*(C*a^3 + 3*A*a^2*b)*m^8 + 6014
254059*(C*a^3 + 3*A*a^2*b)*m^7 + 36588367376*(C*a^3 + 3*A*a^2*b)*m^6 + 163038108552*(C*a^3 + 3*A*a^2*b)*m^5 +
520557781424*(C*a^3 + 3*A*a^2*b)*m^4 + 435891456000*C*a^3 + 1307674368000*A*a^2*b + 1145140001328*(C*a^3 + 3*A
*a^2*b)*m^3 + 1621575699840*(C*a^3 + 3*A*a^2*b)*m^2 + 1301090515200*(C*a^3 + 3*A*a^2*b)*m)*x^3 + (B*a^3*m^14 +
118*B*a^3*m^13 + 6344*B*a^3*m^12 + 205712*B*a^3*m^11 + 4488198*B*a^3*m^10 + 69582084*B*a^3*m^9 + 788931572*B*
a^3*m^8 + 6629764856*B*a^3*m^7 + 41371599841*B*a^3*m^6 + 190060010998*B*a^3*m^5 + 629552085084*B*a^3*m^4 + 144
7709175432*B*a^3*m^3 + 2161577352960*B*a^3*m^2 + 1842662908800*B*a^3*m + 653837184000*B*a^3)*x^2 + (A*a^3*m^14
+ 119*A*a^3*m^13 + 6461*A*a^3*m^12 + 211939*A*a^3*m^11 + 4687683*A*a^3*m^10 + 73870797*A*a^3*m^9 + 854224943*
A*a^3*m^8 + 7353403057*A*a^3*m^7 + 47277726496*A*a^3*m^6 + 225525484184*A*a^3*m^5 + 784146622896*A*a^3*m^4 + 1
922666722704*A*a^3*m^3 + 3134328981120*A*a^3*m^2 + 3031488633600*A*a^3*m + 1307674368000*A*a^3)*x)*(d*x)^m/(m^
15 + 120*m^14 + 6580*m^13 + 218400*m^12 + 4899622*m^11 + 78558480*m^10 + 928095740*m^9 + 8207628000*m^8 + 5463
1129553*m^7 + 272803210680*m^6 + 1009672107080*m^5 + 2706813345600*m^4 + 5056995703824*m^3 + 6165817614720*m^2
+ 4339163001600*m + 1307674368000)

________________________________________________________________________________________

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((d*x)**m*(C*x**2+B*x+A)*(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.33832, size = 10541, normalized size = 26.42 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(C*x^2+B*x+A)*(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

((d*x)^m*C*c^3*m^14*x^15 + (d*x)^m*B*c^3*m^14*x^14 + 105*(d*x)^m*C*c^3*m^13*x^15 + 3*(d*x)^m*C*b*c^2*m^14*x^13
+ (d*x)^m*A*c^3*m^14*x^13 + 106*(d*x)^m*B*c^3*m^13*x^14 + 5005*(d*x)^m*C*c^3*m^12*x^15 + 3*(d*x)^m*B*b*c^2*m^
14*x^12 + 321*(d*x)^m*C*b*c^2*m^13*x^13 + 107*(d*x)^m*A*c^3*m^13*x^13 + 5096*(d*x)^m*B*c^3*m^12*x^14 + 143325*
(d*x)^m*C*c^3*m^11*x^15 + 3*(d*x)^m*C*b^2*c*m^14*x^11 + 3*(d*x)^m*C*a*c^2*m^14*x^11 + 3*(d*x)^m*A*b*c^2*m^14*x
^11 + 324*(d*x)^m*B*b*c^2*m^13*x^12 + 15567*(d*x)^m*C*b*c^2*m^12*x^13 + 5189*(d*x)^m*A*c^3*m^12*x^13 + 147056*
(d*x)^m*B*c^3*m^11*x^14 + 2749747*(d*x)^m*C*c^3*m^10*x^15 + 3*(d*x)^m*B*b^2*c*m^14*x^10 + 3*(d*x)^m*B*a*c^2*m^
14*x^10 + 327*(d*x)^m*C*b^2*c*m^13*x^11 + 327*(d*x)^m*C*a*c^2*m^13*x^11 + 327*(d*x)^m*A*b*c^2*m^13*x^11 + 1585
2*(d*x)^m*B*b*c^2*m^12*x^12 + 452829*(d*x)^m*C*b*c^2*m^11*x^13 + 150943*(d*x)^m*A*c^3*m^11*x^13 + 2840838*(d*x
)^m*B*c^3*m^10*x^14 + 37312275*(d*x)^m*C*c^3*m^9*x^15 + (d*x)^m*C*b^3*m^14*x^9 + 6*(d*x)^m*C*a*b*c*m^14*x^9 +
3*(d*x)^m*A*b^2*c*m^14*x^9 + 3*(d*x)^m*A*a*c^2*m^14*x^9 + 330*(d*x)^m*B*b^2*c*m^13*x^10 + 330*(d*x)^m*B*a*c^2*
m^13*x^10 + 16143*(d*x)^m*C*b^2*c*m^12*x^11 + 16143*(d*x)^m*C*a*c^2*m^12*x^11 + 16143*(d*x)^m*A*b*c^2*m^12*x^1
1 + 464976*(d*x)^m*B*b*c^2*m^11*x^12 + 8812089*(d*x)^m*C*b*c^2*m^10*x^13 + 2937363*(d*x)^m*A*c^3*m^10*x^13 + 3
8786748*(d*x)^m*B*c^3*m^9*x^14 + 368411615*(d*x)^m*C*c^3*m^8*x^15 + (d*x)^m*B*b^3*m^14*x^8 + 6*(d*x)^m*B*a*b*c
*m^14*x^8 + 111*(d*x)^m*C*b^3*m^13*x^9 + 666*(d*x)^m*C*a*b*c*m^13*x^9 + 333*(d*x)^m*A*b^2*c*m^13*x^9 + 333*(d*
x)^m*A*a*c^2*m^13*x^9 + 16440*(d*x)^m*B*b^2*c*m^12*x^10 + 16440*(d*x)^m*B*a*c^2*m^12*x^10 + 477627*(d*x)^m*C*b
^2*c*m^11*x^11 + 477627*(d*x)^m*C*a*c^2*m^11*x^11 + 477627*(d*x)^m*A*b*c^2*m^11*x^11 + 9119154*(d*x)^m*B*b*c^2
*m^10*x^12 + 121118283*(d*x)^m*C*b*c^2*m^9*x^13 + 40372761*(d*x)^m*A*c^3*m^9*x^13 + 385081268*(d*x)^m*B*c^3*m^
8*x^14 + 2681453775*(d*x)^m*C*c^3*m^7*x^15 + 3*(d*x)^m*C*a*b^2*m^14*x^7 + (d*x)^m*A*b^3*m^14*x^7 + 3*(d*x)^m*C
*a^2*c*m^14*x^7 + 6*(d*x)^m*A*a*b*c*m^14*x^7 + 112*(d*x)^m*B*b^3*m^13*x^8 + 672*(d*x)^m*B*a*b*c*m^13*x^8 + 558
1*(d*x)^m*C*b^3*m^12*x^9 + 33486*(d*x)^m*C*a*b*c*m^12*x^9 + 16743*(d*x)^m*A*b^2*c*m^12*x^9 + 16743*(d*x)^m*A*a
*c^2*m^12*x^9 + 490800*(d*x)^m*B*b^2*c*m^11*x^10 + 490800*(d*x)^m*B*a*c^2*m^11*x^10 + 9444969*(d*x)^m*C*b^2*c*
m^10*x^11 + 9444969*(d*x)^m*C*a*c^2*m^10*x^11 + 9444969*(d*x)^m*A*b*c^2*m^10*x^11 + 126245592*(d*x)^m*B*b*c^2*
m^9*x^12 + 1209749541*(d*x)^m*C*b*c^2*m^8*x^13 + 403249847*(d*x)^m*A*c^3*m^8*x^13 + 2816490248*(d*x)^m*B*c^3*m
^7*x^14 + 14409322928*(d*x)^m*C*c^3*m^6*x^15 + 3*(d*x)^m*B*a*b^2*m^14*x^6 + 3*(d*x)^m*B*a^2*c*m^14*x^6 + 339*(
d*x)^m*C*a*b^2*m^13*x^7 + 113*(d*x)^m*A*b^3*m^13*x^7 + 339*(d*x)^m*C*a^2*c*m^13*x^7 + 678*(d*x)^m*A*a*b*c*m^13
*x^7 + 5684*(d*x)^m*B*b^3*m^12*x^8 + 34104*(d*x)^m*B*a*b*c*m^12*x^8 + 168171*(d*x)^m*C*b^3*m^11*x^9 + 1009026*
(d*x)^m*C*a*b*c*m^11*x^9 + 504513*(d*x)^m*A*b^2*c*m^11*x^9 + 504513*(d*x)^m*A*a*c^2*m^11*x^9 + 9790866*(d*x)^m
*B*b^2*c*m^10*x^10 + 9790866*(d*x)^m*B*a*c^2*m^10*x^10 + 131780781*(d*x)^m*C*b^2*c*m^9*x^11 + 131780781*(d*x)^
m*C*a*c^2*m^9*x^11 + 131780781*(d*x)^m*A*b*c^2*m^9*x^11 + 1269340116*(d*x)^m*B*b*c^2*m^8*x^12 + 8896139967*(d*
x)^m*C*b*c^2*m^7*x^13 + 2965379989*(d*x)^m*A*c^3*m^7*x^13 + 15200266081*(d*x)^m*B*c^3*m^6*x^14 + 56663366760*(
d*x)^m*C*c^3*m^5*x^15 + 3*(d*x)^m*C*a^2*b*m^14*x^5 + 3*(d*x)^m*A*a*b^2*m^14*x^5 + 3*(d*x)^m*A*a^2*c*m^14*x^5 +
342*(d*x)^m*B*a*b^2*m^13*x^6 + 342*(d*x)^m*B*a^2*c*m^13*x^6 + 17367*(d*x)^m*C*a*b^2*m^12*x^7 + 5789*(d*x)^m*A
*b^3*m^12*x^7 + 17367*(d*x)^m*C*a^2*c*m^12*x^7 + 34734*(d*x)^m*A*a*b*c*m^12*x^7 + 172928*(d*x)^m*B*b^3*m^11*x^
8 + 1037568*(d*x)^m*B*a*b*c*m^11*x^8 + 3386083*(d*x)^m*C*b^3*m^10*x^9 + 20316498*(d*x)^m*C*a*b*c*m^10*x^9 + 10
158249*(d*x)^m*A*b^2*c*m^10*x^9 + 10158249*(d*x)^m*A*a*c^2*m^10*x^9 + 137766780*(d*x)^m*B*b^2*c*m^9*x^10 + 137
766780*(d*x)^m*B*a*c^2*m^9*x^10 + 1334698629*(d*x)^m*C*b^2*c*m^8*x^11 + 1334698629*(d*x)^m*C*a*c^2*m^8*x^11 +
1334698629*(d*x)^m*A*b*c^2*m^8*x^11 + 9390802608*(d*x)^m*B*b*c^2*m^7*x^12 + 48243569088*(d*x)^m*C*b*c^2*m^6*x^
13 + 16081189696*(d*x)^m*A*c^3*m^6*x^13 + 59999485546*(d*x)^m*B*c^3*m^5*x^14 + 159721605680*(d*x)^m*C*c^3*m^4*
x^15 + 3*(d*x)^m*B*a^2*b*m^14*x^4 + 345*(d*x)^m*C*a^2*b*m^13*x^5 + 345*(d*x)^m*A*a*b^2*m^13*x^5 + 345*(d*x)^m*
A*a^2*c*m^13*x^5 + 17688*(d*x)^m*B*a*b^2*m^12*x^6 + 17688*(d*x)^m*B*a^2*c*m^12*x^6 + 533631*(d*x)^m*C*a*b^2*m^
11*x^7 + 177877*(d*x)^m*A*b^3*m^11*x^7 + 533631*(d*x)^m*C*a^2*c*m^11*x^7 + 1067262*(d*x)^m*A*a*b*c*m^11*x^7 +
3516198*(d*x)^m*B*b^3*m^10*x^8 + 21097188*(d*x)^m*B*a*b*c*m^10*x^8 + 48083733*(d*x)^m*C*b^3*m^9*x^9 + 28850239
8*(d*x)^m*C*a*b*c*m^9*x^9 + 144251199*(d*x)^m*A*b^2*c*m^9*x^9 + 144251199*(d*x)^m*A*a*c^2*m^9*x^9 + 1406619420
*(d*x)^m*B*b^2*c*m^8*x^10 + 1406619420*(d*x)^m*B*a*c^2*m^8*x^10 + 9941199081*(d*x)^m*C*b^2*c*m^7*x^11 + 994119
9081*(d*x)^m*C*a*c^2*m^7*x^11 + 9941199081*(d*x)^m*A*b*c^2*m^7*x^11 + 51203757363*(d*x)^m*B*b*c^2*m^6*x^12 + 1
91243233896*(d*x)^m*C*b*c^2*m^5*x^13 + 63747744632*(d*x)^m*A*c^3*m^5*x^13 + 169679309436*(d*x)^m*B*c^3*m^4*x^1
4 + 310989260400*(d*x)^m*C*c^3*m^3*x^15 + (d*x)^m*C*a^3*m^14*x^3 + 3*(d*x)^m*A*a^2*b*m^14*x^3 + 348*(d*x)^m*B*
a^2*b*m^13*x^4 + 18015*(d*x)^m*C*a^2*b*m^12*x^5 + 18015*(d*x)^m*A*a*b^2*m^12*x^5 + 18015*(d*x)^m*A*a^2*c*m^12*
x^5 + 549072*(d*x)^m*B*a*b^2*m^11*x^6 + 549072*(d*x)^m*B*a^2*c*m^11*x^6 + 10963449*(d*x)^m*C*a*b^2*m^10*x^7 +
3654483*(d*x)^m*A*b^3*m^10*x^7 + 10963449*(d*x)^m*C*a^2*c*m^10*x^7 + 21926898*(d*x)^m*A*a*b*c*m^10*x^7 + 50428
896*(d*x)^m*B*b^3*m^9*x^8 + 302573376*(d*x)^m*B*a*b*c*m^9*x^8 + 495342143*(d*x)^m*C*b^3*m^8*x^9 + 2972052858*(
d*x)^m*C*a*b*c*m^8*x^9 + 1486026429*(d*x)^m*A*b^2*c*m^8*x^9 + 1486026429*(d*x)^m*A*a*c^2*m^8*x^9 + 10556689800
*(d*x)^m*B*b^2*c*m^7*x^10 + 10556689800*(d*x)^m*B*a*c^2*m^7*x^10 + 54540198768*(d*x)^m*C*b^2*c*m^6*x^11 + 5454
0198768*(d*x)^m*C*a*c^2*m^6*x^11 + 54540198768*(d*x)^m*A*b*c^2*m^6*x^11 + 203964543684*(d*x)^m*B*b*c^2*m^5*x^1
2 + 542854280592*(d*x)^m*C*b*c^2*m^4*x^13 + 180951426864*(d*x)^m*A*c^3*m^4*x^13 + 331303013496*(d*x)^m*B*c^3*m
^3*x^14 + 392156797824*(d*x)^m*C*c^3*m^2*x^15 + (d*x)^m*B*a^3*m^14*x^2 + 117*(d*x)^m*C*a^3*m^13*x^3 + 351*(d*x
)^m*A*a^2*b*m^13*x^3 + 18348*(d*x)^m*B*a^2*b*m^12*x^4 + 565125*(d*x)^m*C*a^2*b*m^11*x^5 + 565125*(d*x)^m*A*a*b
^2*m^11*x^5 + 565125*(d*x)^m*A*a^2*c*m^11*x^5 + 11404434*(d*x)^m*B*a*b^2*m^10*x^6 + 11404434*(d*x)^m*B*a^2*c*m
^10*x^6 + 158931297*(d*x)^m*C*a*b^2*m^9*x^7 + 52977099*(d*x)^m*A*b^3*m^9*x^7 + 158931297*(d*x)^m*C*a^2*c*m^9*x
^7 + 317862594*(d*x)^m*A*a*b*c*m^9*x^7 + 524664572*(d*x)^m*B*b^3*m^8*x^8 + 3147987432*(d*x)^m*B*a*b*c*m^8*x^8
+ 3749548713*(d*x)^m*C*b^3*m^7*x^9 + 22497292278*(d*x)^m*C*a*b*c*m^7*x^9 + 11248646139*(d*x)^m*A*b^2*c*m^7*x^9
+ 11248646139*(d*x)^m*A*a*c^2*m^7*x^9 + 58326490659*(d*x)^m*B*b^2*c*m^6*x^10 + 58326490659*(d*x)^m*B*a*c^2*m^
6*x^10 + 218467445592*(d*x)^m*C*b^2*c*m^5*x^11 + 218467445592*(d*x)^m*C*a*c^2*m^5*x^11 + 218467445592*(d*x)^m*
A*b*c^2*m^5*x^11 + 581441797032*(d*x)^m*B*b*c^2*m^4*x^12 + 1063334389104*(d*x)^m*C*b*c^2*m^3*x^13 + 3544447963
68*(d*x)^m*A*c^3*m^3*x^13 + 418753514880*(d*x)^m*B*c^3*m^2*x^14 + 283465647360*(d*x)^m*C*c^3*m*x^15 + (d*x)^m*
A*a^3*m^14*x + 118*(d*x)^m*B*a^3*m^13*x^2 + 6229*(d*x)^m*C*a^3*m^12*x^3 + 18687*(d*x)^m*A*a^2*b*m^12*x^3 + 581
808*(d*x)^m*B*a^2*b*m^11*x^4 + 11873241*(d*x)^m*C*a^2*b*m^10*x^5 + 11873241*(d*x)^m*A*a*b^2*m^10*x^5 + 1187324
1*(d*x)^m*A*a^2*c*m^10*x^5 + 167248836*(d*x)^m*B*a*b^2*m^9*x^6 + 167248836*(d*x)^m*B*a^2*c*m^9*x^6 + 167176814
1*(d*x)^m*C*a*b^2*m^8*x^7 + 557256047*(d*x)^m*A*b^3*m^8*x^7 + 1671768141*(d*x)^m*C*a^2*c*m^8*x^7 + 3343536282*
(d*x)^m*A*a*b*c*m^8*x^7 + 4010311424*(d*x)^m*B*b^3*m^7*x^8 + 24061868544*(d*x)^m*B*a*b*c*m^7*x^8 + 20885191136
*(d*x)^m*C*b^3*m^6*x^9 + 125311146816*(d*x)^m*C*a*b*c*m^6*x^9 + 62655573408*(d*x)^m*A*b^2*c*m^6*x^9 + 62655573
408*(d*x)^m*A*a*c^2*m^6*x^9 + 235144725450*(d*x)^m*B*b^2*c*m^5*x^10 + 235144725450*(d*x)^m*B*a*c^2*m^5*x^10 +
625874419728*(d*x)^m*C*b^2*c*m^4*x^11 + 625874419728*(d*x)^m*C*a*c^2*m^4*x^11 + 625874419728*(d*x)^m*A*b*c^2*m
^4*x^11 + 1143138472416*(d*x)^m*B*b*c^2*m^3*x^12 + 1347640053120*(d*x)^m*C*b*c^2*m^2*x^13 + 449213351040*(d*x)
^m*A*c^3*m^2*x^13 + 303268406400*(d*x)^m*B*c^3*m*x^14 + 87178291200*(d*x)^m*C*c^3*x^15 + 119*(d*x)^m*A*a^3*m^1
3*x + 6344*(d*x)^m*B*a^3*m^12*x^2 + 199713*(d*x)^m*C*a^3*m^11*x^3 + 599139*(d*x)^m*A*a^2*b*m^11*x^3 + 12371634
*(d*x)^m*B*a^2*b*m^10*x^4 + 176309235*(d*x)^m*C*a^2*b*m^9*x^5 + 176309235*(d*x)^m*A*a*b^2*m^9*x^5 + 176309235*
(d*x)^m*A*a^2*c*m^9*x^5 + 1780794204*(d*x)^m*B*a*b^2*m^8*x^6 + 1780794204*(d*x)^m*B*a^2*c*m^8*x^6 + 1292050701
3*(d*x)^m*C*a*b^2*m^7*x^7 + 4306835671*(d*x)^m*A*b^3*m^7*x^7 + 12920507013*(d*x)^m*C*a^2*c*m^7*x^7 + 258410140
26*(d*x)^m*A*a*b*c*m^7*x^7 + 22548638161*(d*x)^m*B*b^3*m^6*x^8 + 135291828966*(d*x)^m*B*a*b*c*m^6*x^8 + 848364
90456*(d*x)^m*C*b^3*m^5*x^9 + 509018942736*(d*x)^m*C*a*b*c*m^5*x^9 + 254509471368*(d*x)^m*A*b^2*c*m^5*x^9 + 25
4509471368*(d*x)^m*A*a*c^2*m^5*x^9 + 677569066740*(d*x)^m*B*b^2*c*m^4*x^10 + 677569066740*(d*x)^m*B*a*c^2*m^4*
x^10 + 1235821419792*(d*x)^m*C*b^2*c*m^3*x^11 + 1235821419792*(d*x)^m*C*a*c^2*m^3*x^11 + 1235821419792*(d*x)^m
*A*b*c^2*m^3*x^11 + 1453325442480*(d*x)^m*B*b*c^2*m^2*x^12 + 978132153600*(d*x)^m*C*b*c^2*m*x^13 + 32604405120
0*(d*x)^m*A*c^3*m*x^13 + 93405312000*(d*x)^m*B*c^3*x^14 + 6461*(d*x)^m*A*a^3*m^12*x + 205712*(d*x)^m*B*a^3*m^1
1*x^2 + 4300483*(d*x)^m*C*a^3*m^10*x^3 + 12901449*(d*x)^m*A*a^2*b*m^10*x^3 + 186188904*(d*x)^m*B*a^2*b*m^9*x^4
+ 1902741045*(d*x)^m*C*a^2*b*m^8*x^5 + 1902741045*(d*x)^m*A*a*b^2*m^8*x^5 + 1902741045*(d*x)^m*A*a^2*c*m^8*x^
5 + 13938118776*(d*x)^m*B*a*b^2*m^7*x^6 + 13938118776*(d*x)^m*B*a^2*c*m^7*x^6 + 73449839568*(d*x)^m*C*a*b^2*m^
6*x^7 + 24483279856*(d*x)^m*A*b^3*m^6*x^7 + 73449839568*(d*x)^m*C*a^2*c*m^6*x^7 + 146899679136*(d*x)^m*A*a*b*c
*m^6*x^7 + 92414105392*(d*x)^m*B*b^3*m^5*x^8 + 554484632352*(d*x)^m*B*a*b*c*m^5*x^8 + 246143692976*(d*x)^m*C*b
^3*m^4*x^9 + 1476862157856*(d*x)^m*C*a*b*c*m^4*x^9 + 738431078928*(d*x)^m*A*b^2*c*m^4*x^9 + 738431078928*(d*x)
^m*A*a*c^2*m^4*x^9 + 1344749369400*(d*x)^m*B*b^2*c*m^3*x^10 + 1344749369400*(d*x)^m*B*a*c^2*m^3*x^10 + 1576951
493760*(d*x)^m*C*b^2*c*m^2*x^11 + 1576951493760*(d*x)^m*C*a*c^2*m^2*x^11 + 1576951493760*(d*x)^m*A*b*c^2*m^2*x
^11 + 1057547534400*(d*x)^m*B*b*c^2*m*x^12 + 301771008000*(d*x)^m*C*b*c^2*x^13 + 100590336000*(d*x)^m*A*c^3*x^
13 + 211939*(d*x)^m*A*a^3*m^11*x + 4488198*(d*x)^m*B*a^3*m^10*x^2 + 65657031*(d*x)^m*C*a^3*m^9*x^3 + 196971093
*(d*x)^m*A*a^2*b*m^9*x^3 + 2039531604*(d*x)^m*B*a^2*b*m^8*x^4 + 15109178775*(d*x)^m*C*a^2*b*m^7*x^5 + 15109178
775*(d*x)^m*A*a*b^2*m^7*x^5 + 15109178775*(d*x)^m*A*a^2*c*m^7*x^5 + 80264676003*(d*x)^m*B*a*b^2*m^6*x^6 + 8026
4676003*(d*x)^m*B*a^2*c*m^6*x^6 + 304260755064*(d*x)^m*C*a*b^2*m^5*x^7 + 101420251688*(d*x)^m*A*b^3*m^5*x^7 +
304260755064*(d*x)^m*C*a^2*c*m^5*x^7 + 608521510128*(d*x)^m*A*a*b*c*m^5*x^7 + 270359263944*(d*x)^m*B*b^3*m^4*x
^8 + 1622155583664*(d*x)^m*B*a*b*c*m^4*x^8 + 491520108816*(d*x)^m*C*b^3*m^3*x^9 + 2949120652896*(d*x)^m*C*a*b*
c*m^3*x^9 + 1474560326448*(d*x)^m*A*b^2*c*m^3*x^9 + 1474560326448*(d*x)^m*A*a*c^2*m^3*x^9 + 1723493417472*(d*x
)^m*B*b^2*c*m^2*x^10 + 1723493417472*(d*x)^m*B*a*c^2*m^2*x^10 + 1150986412800*(d*x)^m*C*b^2*c*m*x^11 + 1150986
412800*(d*x)^m*C*a*c^2*m*x^11 + 1150986412800*(d*x)^m*A*b*c^2*m*x^11 + 326918592000*(d*x)^m*B*b*c^2*x^12 + 468
7683*(d*x)^m*A*a^3*m^10*x + 69582084*(d*x)^m*B*a^3*m^9*x^2 + 731124647*(d*x)^m*C*a^3*m^8*x^3 + 2193373941*(d*x
)^m*A*a^2*b*m^8*x^3 + 16464757584*(d*x)^m*B*a^2*b*m^7*x^4 + 88347494784*(d*x)^m*C*a^2*b*m^6*x^5 + 88347494784*
(d*x)^m*A*a*b^2*m^6*x^5 + 88347494784*(d*x)^m*A*a^2*c*m^6*x^5 + 336821576022*(d*x)^m*B*a*b^2*m^5*x^6 + 3368215
76022*(d*x)^m*B*a^2*c*m^5*x^6 + 899191035792*(d*x)^m*C*a*b^2*m^4*x^7 + 299730345264*(d*x)^m*A*b^3*m^4*x^7 + 89
9191035792*(d*x)^m*C*a^2*c*m^4*x^7 + 1798382071584*(d*x)^m*A*a*b*c*m^4*x^7 + 543939234048*(d*x)^m*B*b^3*m^3*x^
8 + 3263635404288*(d*x)^m*B*a*b*c*m^3*x^8 + 633314724480*(d*x)^m*C*b^3*m^2*x^9 + 3799888346880*(d*x)^m*C*a*b*c
*m^2*x^9 + 1899944173440*(d*x)^m*A*b^2*c*m^2*x^9 + 1899944173440*(d*x)^m*A*a*c^2*m^2*x^9 + 1262518669440*(d*x)
^m*B*b^2*c*m*x^10 + 1262518669440*(d*x)^m*B*a*c^2*m*x^10 + 356638464000*(d*x)^m*C*b^2*c*x^11 + 356638464000*(d
*x)^m*C*a*c^2*x^11 + 356638464000*(d*x)^m*A*b*c^2*x^11 + 73870797*(d*x)^m*A*a^3*m^9*x + 788931572*(d*x)^m*B*a^
3*m^8*x^2 + 6014254059*(d*x)^m*C*a^3*m^7*x^3 + 18042762177*(d*x)^m*A*a^2*b*m^7*x^3 + 98034358323*(d*x)^m*B*a^2
*b*m^6*x^4 + 376672158120*(d*x)^m*C*a^2*b*m^5*x^5 + 376672158120*(d*x)^m*A*a*b^2*m^5*x^5 + 376672158120*(d*x)^
m*A*a^2*c*m^5*x^5 + 1008086865108*(d*x)^m*B*a*b^2*m^4*x^6 + 1008086865108*(d*x)^m*B*a^2*c*m^4*x^6 + 1826102786
256*(d*x)^m*C*a*b^2*m^3*x^7 + 608700928752*(d*x)^m*A*b^3*m^3*x^7 + 1826102786256*(d*x)^m*C*a^2*c*m^3*x^7 + 365
2205572512*(d*x)^m*A*a*b*c*m^3*x^7 + 705481831440*(d*x)^m*B*b^3*m^2*x^8 + 4232890988640*(d*x)^m*B*a*b*c*m^2*x^
8 + 465985094400*(d*x)^m*C*b^3*m*x^9 + 2795910566400*(d*x)^m*C*a*b*c*m*x^9 + 1397955283200*(d*x)^m*A*b^2*c*m*x
^9 + 1397955283200*(d*x)^m*A*a*c^2*m*x^9 + 392302310400*(d*x)^m*B*b^2*c*x^10 + 392302310400*(d*x)^m*B*a*c^2*x^
10 + 854224943*(d*x)^m*A*a^3*m^8*x + 6629764856*(d*x)^m*B*a^3*m^7*x^2 + 36588367376*(d*x)^m*C*a^3*m^6*x^3 + 10
9765102128*(d*x)^m*A*a^2*b*m^6*x^3 + 426272198748*(d*x)^m*B*a^2*b*m^5*x^4 + 1145655530640*(d*x)^m*C*a^2*b*m^4*
x^5 + 1145655530640*(d*x)^m*A*a*b^2*m^4*x^5 + 1145655530640*(d*x)^m*A*a^2*c*m^4*x^5 + 2071918846152*(d*x)^m*B*
a*b^2*m^3*x^6 + 2071918846152*(d*x)^m*B*a^2*c*m^3*x^6 + 2388267607680*(d*x)^m*C*a*b^2*m^2*x^7 + 796089202560*(
d*x)^m*A*b^3*m^2*x^7 + 2388267607680*(d*x)^m*C*a^2*c*m^2*x^7 + 4776535215360*(d*x)^m*A*a*b*c*m^2*x^7 + 5219629
63200*(d*x)^m*B*b^3*m*x^8 + 3131777779200*(d*x)^m*B*a*b*c*m*x^8 + 145297152000*(d*x)^m*C*b^3*x^9 + 87178291200
0*(d*x)^m*C*a*b*c*x^9 + 435891456000*(d*x)^m*A*b^2*c*x^9 + 435891456000*(d*x)^m*A*a*c^2*x^9 + 7353403057*(d*x)
^m*A*a^3*m^7*x + 41371599841*(d*x)^m*B*a^3*m^6*x^2 + 163038108552*(d*x)^m*C*a^3*m^5*x^3 + 489114325656*(d*x)^m
*A*a^2*b*m^5*x^3 + 1323927526248*(d*x)^m*B*a^2*b*m^4*x^4 + 2392162383600*(d*x)^m*C*a^2*b*m^3*x^5 + 23921623836
00*(d*x)^m*A*a*b^2*m^3*x^5 + 2392162383600*(d*x)^m*A*a^2*c*m^3*x^5 + 2739474034560*(d*x)^m*B*a*b^2*m^2*x^6 + 2
739474034560*(d*x)^m*B*a^2*c*m^2*x^6 + 1779579590400*(d*x)^m*C*a*b^2*m*x^7 + 593193196800*(d*x)^m*A*b^3*m*x^7
+ 1779579590400*(d*x)^m*C*a^2*c*m*x^7 + 3559159180800*(d*x)^m*A*a*b*c*m*x^7 + 163459296000*(d*x)^m*B*b^3*x^8 +
980755776000*(d*x)^m*B*a*b*c*x^8 + 47277726496*(d*x)^m*A*a^3*m^6*x + 190060010998*(d*x)^m*B*a^3*m^5*x^2 + 520
557781424*(d*x)^m*C*a^3*m^4*x^3 + 1561673344272*(d*x)^m*A*a^2*b*m^4*x^3 + 2824729931808*(d*x)^m*B*a^2*b*m^3*x^
4 + 3210175193472*(d*x)^m*C*a^2*b*m^2*x^5 + 3210175193472*(d*x)^m*A*a*b^2*m^2*x^5 + 3210175193472*(d*x)^m*A*a^
2*c*m^2*x^5 + 2060608636800*(d*x)^m*B*a*b^2*m*x^6 + 2060608636800*(d*x)^m*B*a^2*c*m*x^6 + 560431872000*(d*x)^m
*C*a*b^2*x^7 + 186810624000*(d*x)^m*A*b^3*x^7 + 560431872000*(d*x)^m*C*a^2*c*x^7 + 1120863744000*(d*x)^m*A*a*b
*c*x^7 + 225525484184*(d*x)^m*A*a^3*m^5*x + 629552085084*(d*x)^m*B*a^3*m^4*x^2 + 1145140001328*(d*x)^m*C*a^3*m
^3*x^3 + 3435420003984*(d*x)^m*A*a^2*b*m^3*x^3 + 3872067384240*(d*x)^m*B*a^2*b*m^2*x^4 + 2446576876800*(d*x)^m
*C*a^2*b*m*x^5 + 2446576876800*(d*x)^m*A*a*b^2*m*x^5 + 2446576876800*(d*x)^m*A*a^2*c*m*x^5 + 653837184000*(d*x
)^m*B*a*b^2*x^6 + 653837184000*(d*x)^m*B*a^2*c*x^6 + 784146622896*(d*x)^m*A*a^3*m^4*x + 1447709175432*(d*x)^m*
B*a^3*m^3*x^2 + 1621575699840*(d*x)^m*C*a^3*m^2*x^3 + 4864727099520*(d*x)^m*A*a^2*b*m^2*x^3 + 3009183307200*(d
*x)^m*B*a^2*b*m*x^4 + 784604620800*(d*x)^m*C*a^2*b*x^5 + 784604620800*(d*x)^m*A*a*b^2*x^5 + 784604620800*(d*x)
^m*A*a^2*c*x^5 + 1922666722704*(d*x)^m*A*a^3*m^3*x + 2161577352960*(d*x)^m*B*a^3*m^2*x^2 + 1301090515200*(d*x)
^m*C*a^3*m*x^3 + 3903271545600*(d*x)^m*A*a^2*b*m*x^3 + 980755776000*(d*x)^m*B*a^2*b*x^4 + 3134328981120*(d*x)^
m*A*a^3*m^2*x + 1842662908800*(d*x)^m*B*a^3*m*x^2 + 435891456000*(d*x)^m*C*a^3*x^3 + 1307674368000*(d*x)^m*A*a
^2*b*x^3 + 3031488633600*(d*x)^m*A*a^3*m*x + 653837184000*(d*x)^m*B*a^3*x^2 + 1307674368000*(d*x)^m*A*a^3*x)/(
m^15 + 120*m^14 + 6580*m^13 + 218400*m^12 + 4899622*m^11 + 78558480*m^10 + 928095740*m^9 + 8207628000*m^8 + 54
631129553*m^7 + 272803210680*m^6 + 1009672107080*m^5 + 2706813345600*m^4 + 5056995703824*m^3 + 6165817614720*m
^2 + 4339163001600*m + 1307674368000)