3.269 \(\int x^m (A+B x^2) (b x^2+c x^4)^3 \, dx\)

Optimal. Leaf size=96 \[ \frac{b^2 x^{m+9} (3 A c+b B)}{m+9}+\frac{A b^3 x^{m+7}}{m+7}+\frac{c^2 x^{m+13} (A c+3 b B)}{m+13}+\frac{3 b c x^{m+11} (A c+b B)}{m+11}+\frac{B c^3 x^{m+15}}{m+15} \]

[Out]

(A*b^3*x^(7 + m))/(7 + m) + (b^2*(b*B + 3*A*c)*x^(9 + m))/(9 + m) + (3*b*c*(b*B + A*c)*x^(11 + m))/(11 + m) +
(c^2*(3*b*B + A*c)*x^(13 + m))/(13 + m) + (B*c^3*x^(15 + m))/(15 + m)

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Rubi [A]  time = 0.0699623, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1584, 448} \[ \frac{b^2 x^{m+9} (3 A c+b B)}{m+9}+\frac{A b^3 x^{m+7}}{m+7}+\frac{c^2 x^{m+13} (A c+3 b B)}{m+13}+\frac{3 b c x^{m+11} (A c+b B)}{m+11}+\frac{B c^3 x^{m+15}}{m+15} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^3*x^(7 + m))/(7 + m) + (b^2*(b*B + 3*A*c)*x^(9 + m))/(9 + m) + (3*b*c*(b*B + A*c)*x^(11 + m))/(11 + m) +
(c^2*(3*b*B + A*c)*x^(13 + m))/(13 + m) + (B*c^3*x^(15 + m))/(15 + m)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

Mathematica [A]  time = 0.091216, size = 89, normalized size = 0.93 \[ x^{m+7} \left (\frac{b^2 x^2 (3 A c+b B)}{m+9}+\frac{A b^3}{m+7}+\frac{c^2 x^6 (A c+3 b B)}{m+13}+\frac{3 b c x^4 (A c+b B)}{m+11}+\frac{B c^3 x^8}{m+15}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^(7 + m)*((A*b^3)/(7 + m) + (b^2*(b*B + 3*A*c)*x^2)/(9 + m) + (3*b*c*(b*B + A*c)*x^4)/(11 + m) + (c^2*(3*b*B
+ A*c)*x^6)/(13 + m) + (B*c^3*x^8)/(15 + m))

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Maple [B]  time = 0.051, size = 474, normalized size = 4.9 \begin{align*}{\frac{{x}^{7+m} \left ( B{c}^{3}{m}^{4}{x}^{8}+40\,B{c}^{3}{m}^{3}{x}^{8}+A{c}^{3}{m}^{4}{x}^{6}+3\,Bb{c}^{2}{m}^{4}{x}^{6}+590\,B{c}^{3}{m}^{2}{x}^{8}+42\,A{c}^{3}{m}^{3}{x}^{6}+126\,Bb{c}^{2}{m}^{3}{x}^{6}+3800\,B{c}^{3}m{x}^{8}+3\,Ab{c}^{2}{m}^{4}{x}^{4}+644\,A{c}^{3}{m}^{2}{x}^{6}+3\,B{b}^{2}c{m}^{4}{x}^{4}+1932\,Bb{c}^{2}{m}^{2}{x}^{6}+9009\,B{c}^{3}{x}^{8}+132\,Ab{c}^{2}{m}^{3}{x}^{4}+4278\,A{c}^{3}m{x}^{6}+132\,B{b}^{2}c{m}^{3}{x}^{4}+12834\,Bb{c}^{2}m{x}^{6}+3\,A{b}^{2}c{m}^{4}{x}^{2}+2118\,Ab{c}^{2}{m}^{2}{x}^{4}+10395\,A{c}^{3}{x}^{6}+B{b}^{3}{m}^{4}{x}^{2}+2118\,B{b}^{2}c{m}^{2}{x}^{4}+31185\,B{x}^{6}b{c}^{2}+138\,A{b}^{2}c{m}^{3}{x}^{2}+14652\,Ab{c}^{2}m{x}^{4}+46\,B{b}^{3}{m}^{3}{x}^{2}+14652\,B{b}^{2}cm{x}^{4}+A{b}^{3}{m}^{4}+2328\,A{b}^{2}c{m}^{2}{x}^{2}+36855\,Ab{c}^{2}{x}^{4}+776\,B{b}^{3}{m}^{2}{x}^{2}+36855\,B{x}^{4}{b}^{2}c+48\,A{b}^{3}{m}^{3}+16998\,A{b}^{2}cm{x}^{2}+5666\,B{b}^{3}m{x}^{2}+854\,A{b}^{3}{m}^{2}+45045\,A{b}^{2}c{x}^{2}+15015\,B{x}^{2}{b}^{3}+6672\,A{b}^{3}m+19305\,A{b}^{3} \right ) }{ \left ( 15+m \right ) \left ( 13+m \right ) \left ( 11+m \right ) \left ( 9+m \right ) \left ( 7+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(7+m)*(B*c^3*m^4*x^8+40*B*c^3*m^3*x^8+A*c^3*m^4*x^6+3*B*b*c^2*m^4*x^6+590*B*c^3*m^2*x^8+42*A*c^3*m^3*x^6+126
*B*b*c^2*m^3*x^6+3800*B*c^3*m*x^8+3*A*b*c^2*m^4*x^4+644*A*c^3*m^2*x^6+3*B*b^2*c*m^4*x^4+1932*B*b*c^2*m^2*x^6+9
009*B*c^3*x^8+132*A*b*c^2*m^3*x^4+4278*A*c^3*m*x^6+132*B*b^2*c*m^3*x^4+12834*B*b*c^2*m*x^6+3*A*b^2*c*m^4*x^2+2
118*A*b*c^2*m^2*x^4+10395*A*c^3*x^6+B*b^3*m^4*x^2+2118*B*b^2*c*m^2*x^4+31185*B*b*c^2*x^6+138*A*b^2*c*m^3*x^2+1
4652*A*b*c^2*m*x^4+46*B*b^3*m^3*x^2+14652*B*b^2*c*m*x^4+A*b^3*m^4+2328*A*b^2*c*m^2*x^2+36855*A*b*c^2*x^4+776*B
*b^3*m^2*x^2+36855*B*b^2*c*x^4+48*A*b^3*m^3+16998*A*b^2*c*m*x^2+5666*B*b^3*m*x^2+854*A*b^3*m^2+45045*A*b^2*c*x
^2+15015*B*b^3*x^2+6672*A*b^3*m+19305*A*b^3)/(15+m)/(13+m)/(11+m)/(9+m)/(7+m)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.29343, size = 910, normalized size = 9.48 \begin{align*} \frac{{\left ({\left (B c^{3} m^{4} + 40 \, B c^{3} m^{3} + 590 \, B c^{3} m^{2} + 3800 \, B c^{3} m + 9009 \, B c^{3}\right )} x^{15} +{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} m^{4} + 31185 \, B b c^{2} + 10395 \, A c^{3} + 42 \,{\left (3 \, B b c^{2} + A c^{3}\right )} m^{3} + 644 \,{\left (3 \, B b c^{2} + A c^{3}\right )} m^{2} + 4278 \,{\left (3 \, B b c^{2} + A c^{3}\right )} m\right )} x^{13} + 3 \,{\left ({\left (B b^{2} c + A b c^{2}\right )} m^{4} + 12285 \, B b^{2} c + 12285 \, A b c^{2} + 44 \,{\left (B b^{2} c + A b c^{2}\right )} m^{3} + 706 \,{\left (B b^{2} c + A b c^{2}\right )} m^{2} + 4884 \,{\left (B b^{2} c + A b c^{2}\right )} m\right )} x^{11} +{\left ({\left (B b^{3} + 3 \, A b^{2} c\right )} m^{4} + 15015 \, B b^{3} + 45045 \, A b^{2} c + 46 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} m^{3} + 776 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} m^{2} + 5666 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} m\right )} x^{9} +{\left (A b^{3} m^{4} + 48 \, A b^{3} m^{3} + 854 \, A b^{3} m^{2} + 6672 \, A b^{3} m + 19305 \, A b^{3}\right )} x^{7}\right )} x^{m}}{m^{5} + 55 \, m^{4} + 1190 \, m^{3} + 12650 \, m^{2} + 66009 \, m + 135135} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*c^3*m^4 + 40*B*c^3*m^3 + 590*B*c^3*m^2 + 3800*B*c^3*m + 9009*B*c^3)*x^15 + ((3*B*b*c^2 + A*c^3)*m^4 + 3118
5*B*b*c^2 + 10395*A*c^3 + 42*(3*B*b*c^2 + A*c^3)*m^3 + 644*(3*B*b*c^2 + A*c^3)*m^2 + 4278*(3*B*b*c^2 + A*c^3)*
m)*x^13 + 3*((B*b^2*c + A*b*c^2)*m^4 + 12285*B*b^2*c + 12285*A*b*c^2 + 44*(B*b^2*c + A*b*c^2)*m^3 + 706*(B*b^2
*c + A*b*c^2)*m^2 + 4884*(B*b^2*c + A*b*c^2)*m)*x^11 + ((B*b^3 + 3*A*b^2*c)*m^4 + 15015*B*b^3 + 45045*A*b^2*c
+ 46*(B*b^3 + 3*A*b^2*c)*m^3 + 776*(B*b^3 + 3*A*b^2*c)*m^2 + 5666*(B*b^3 + 3*A*b^2*c)*m)*x^9 + (A*b^3*m^4 + 48
*A*b^3*m^3 + 854*A*b^3*m^2 + 6672*A*b^3*m + 19305*A*b^3)*x^7)*x^m/(m^5 + 55*m^4 + 1190*m^3 + 12650*m^2 + 66009
*m + 135135)

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Sympy [A]  time = 11.2351, size = 2077, normalized size = 21.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(B*x**2+A)*(c*x**4+b*x**2)**3,x)

[Out]

Piecewise((-A*b**3/(8*x**8) - A*b**2*c/(2*x**6) - 3*A*b*c**2/(4*x**4) - A*c**3/(2*x**2) - B*b**3/(6*x**6) - 3*
B*b**2*c/(4*x**4) - 3*B*b*c**2/(2*x**2) + B*c**3*log(x), Eq(m, -15)), (-A*b**3/(6*x**6) - 3*A*b**2*c/(4*x**4)
- 3*A*b*c**2/(2*x**2) + A*c**3*log(x) - B*b**3/(4*x**4) - 3*B*b**2*c/(2*x**2) + 3*B*b*c**2*log(x) + B*c**3*x**
2/2, Eq(m, -13)), (-A*b**3/(4*x**4) - 3*A*b**2*c/(2*x**2) + 3*A*b*c**2*log(x) + A*c**3*x**2/2 - B*b**3/(2*x**2
) + 3*B*b**2*c*log(x) + 3*B*b*c**2*x**2/2 + B*c**3*x**4/4, Eq(m, -11)), (-A*b**3/(2*x**2) + 3*A*b**2*c*log(x)
+ 3*A*b*c**2*x**2/2 + A*c**3*x**4/4 + B*b**3*log(x) + 3*B*b**2*c*x**2/2 + 3*B*b*c**2*x**4/4 + B*c**3*x**6/6, E
q(m, -9)), (A*b**3*log(x) + 3*A*b**2*c*x**2/2 + 3*A*b*c**2*x**4/4 + A*c**3*x**6/6 + B*b**3*x**2/2 + 3*B*b**2*c
*x**4/4 + B*b*c**2*x**6/2 + B*c**3*x**8/8, Eq(m, -7)), (A*b**3*m**4*x**7*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12
650*m**2 + 66009*m + 135135) + 48*A*b**3*m**3*x**7*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 1
35135) + 854*A*b**3*m**2*x**7*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 6672*A*b**3*
m*x**7*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 19305*A*b**3*x**7*x**m/(m**5 + 55*m
**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 3*A*b**2*c*m**4*x**9*x**m/(m**5 + 55*m**4 + 1190*m**3 + 126
50*m**2 + 66009*m + 135135) + 138*A*b**2*c*m**3*x**9*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m +
135135) + 2328*A*b**2*c*m**2*x**9*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 16998*A
*b**2*c*m*x**9*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 45045*A*b**2*c*x**9*x**m/(m
**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 3*A*b*c**2*m**4*x**11*x**m/(m**5 + 55*m**4 + 1190
*m**3 + 12650*m**2 + 66009*m + 135135) + 132*A*b*c**2*m**3*x**11*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2
+ 66009*m + 135135) + 2118*A*b*c**2*m**2*x**11*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 1351
35) + 14652*A*b*c**2*m*x**11*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 36855*A*b*c**
2*x**11*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + A*c**3*m**4*x**13*x**m/(m**5 + 55*
m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 42*A*c**3*m**3*x**13*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12
650*m**2 + 66009*m + 135135) + 644*A*c**3*m**2*x**13*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m +
135135) + 4278*A*c**3*m*x**13*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 10395*A*c**
3*x**13*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + B*b**3*m**4*x**9*x**m/(m**5 + 55*m
**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 46*B*b**3*m**3*x**9*x**m/(m**5 + 55*m**4 + 1190*m**3 + 1265
0*m**2 + 66009*m + 135135) + 776*B*b**3*m**2*x**9*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 13
5135) + 5666*B*b**3*m*x**9*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 15015*B*b**3*x*
*9*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 3*B*b**2*c*m**4*x**11*x**m/(m**5 + 55*m
**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 132*B*b**2*c*m**3*x**11*x**m/(m**5 + 55*m**4 + 1190*m**3 +
12650*m**2 + 66009*m + 135135) + 2118*B*b**2*c*m**2*x**11*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 6600
9*m + 135135) + 14652*B*b**2*c*m*x**11*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 368
55*B*b**2*c*x**11*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 3*B*b*c**2*m**4*x**13*x*
*m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 126*B*b*c**2*m**3*x**13*x**m/(m**5 + 55*m**4
+ 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 1932*B*b*c**2*m**2*x**13*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12
650*m**2 + 66009*m + 135135) + 12834*B*b*c**2*m*x**13*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m
+ 135135) + 31185*B*b*c**2*x**13*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + B*c**3*m*
*4*x**15*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 40*B*c**3*m**3*x**15*x**m/(m**5 +
55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135) + 590*B*c**3*m**2*x**15*x**m/(m**5 + 55*m**4 + 1190*m**3
+ 12650*m**2 + 66009*m + 135135) + 3800*B*c**3*m*x**15*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*
m + 135135) + 9009*B*c**3*x**15*x**m/(m**5 + 55*m**4 + 1190*m**3 + 12650*m**2 + 66009*m + 135135), True))

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Giac [B]  time = 1.29304, size = 814, normalized size = 8.48 \begin{align*} \frac{B c^{3} m^{4} x^{15} x^{m} + 40 \, B c^{3} m^{3} x^{15} x^{m} + 3 \, B b c^{2} m^{4} x^{13} x^{m} + A c^{3} m^{4} x^{13} x^{m} + 590 \, B c^{3} m^{2} x^{15} x^{m} + 126 \, B b c^{2} m^{3} x^{13} x^{m} + 42 \, A c^{3} m^{3} x^{13} x^{m} + 3800 \, B c^{3} m x^{15} x^{m} + 3 \, B b^{2} c m^{4} x^{11} x^{m} + 3 \, A b c^{2} m^{4} x^{11} x^{m} + 1932 \, B b c^{2} m^{2} x^{13} x^{m} + 644 \, A c^{3} m^{2} x^{13} x^{m} + 9009 \, B c^{3} x^{15} x^{m} + 132 \, B b^{2} c m^{3} x^{11} x^{m} + 132 \, A b c^{2} m^{3} x^{11} x^{m} + 12834 \, B b c^{2} m x^{13} x^{m} + 4278 \, A c^{3} m x^{13} x^{m} + B b^{3} m^{4} x^{9} x^{m} + 3 \, A b^{2} c m^{4} x^{9} x^{m} + 2118 \, B b^{2} c m^{2} x^{11} x^{m} + 2118 \, A b c^{2} m^{2} x^{11} x^{m} + 31185 \, B b c^{2} x^{13} x^{m} + 10395 \, A c^{3} x^{13} x^{m} + 46 \, B b^{3} m^{3} x^{9} x^{m} + 138 \, A b^{2} c m^{3} x^{9} x^{m} + 14652 \, B b^{2} c m x^{11} x^{m} + 14652 \, A b c^{2} m x^{11} x^{m} + A b^{3} m^{4} x^{7} x^{m} + 776 \, B b^{3} m^{2} x^{9} x^{m} + 2328 \, A b^{2} c m^{2} x^{9} x^{m} + 36855 \, B b^{2} c x^{11} x^{m} + 36855 \, A b c^{2} x^{11} x^{m} + 48 \, A b^{3} m^{3} x^{7} x^{m} + 5666 \, B b^{3} m x^{9} x^{m} + 16998 \, A b^{2} c m x^{9} x^{m} + 854 \, A b^{3} m^{2} x^{7} x^{m} + 15015 \, B b^{3} x^{9} x^{m} + 45045 \, A b^{2} c x^{9} x^{m} + 6672 \, A b^{3} m x^{7} x^{m} + 19305 \, A b^{3} x^{7} x^{m}}{m^{5} + 55 \, m^{4} + 1190 \, m^{3} + 12650 \, m^{2} + 66009 \, m + 135135} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*c^3*m^4*x^15*x^m + 40*B*c^3*m^3*x^15*x^m + 3*B*b*c^2*m^4*x^13*x^m + A*c^3*m^4*x^13*x^m + 590*B*c^3*m^2*x^15
*x^m + 126*B*b*c^2*m^3*x^13*x^m + 42*A*c^3*m^3*x^13*x^m + 3800*B*c^3*m*x^15*x^m + 3*B*b^2*c*m^4*x^11*x^m + 3*A
*b*c^2*m^4*x^11*x^m + 1932*B*b*c^2*m^2*x^13*x^m + 644*A*c^3*m^2*x^13*x^m + 9009*B*c^3*x^15*x^m + 132*B*b^2*c*m
^3*x^11*x^m + 132*A*b*c^2*m^3*x^11*x^m + 12834*B*b*c^2*m*x^13*x^m + 4278*A*c^3*m*x^13*x^m + B*b^3*m^4*x^9*x^m
+ 3*A*b^2*c*m^4*x^9*x^m + 2118*B*b^2*c*m^2*x^11*x^m + 2118*A*b*c^2*m^2*x^11*x^m + 31185*B*b*c^2*x^13*x^m + 103
95*A*c^3*x^13*x^m + 46*B*b^3*m^3*x^9*x^m + 138*A*b^2*c*m^3*x^9*x^m + 14652*B*b^2*c*m*x^11*x^m + 14652*A*b*c^2*
m*x^11*x^m + A*b^3*m^4*x^7*x^m + 776*B*b^3*m^2*x^9*x^m + 2328*A*b^2*c*m^2*x^9*x^m + 36855*B*b^2*c*x^11*x^m + 3
6855*A*b*c^2*x^11*x^m + 48*A*b^3*m^3*x^7*x^m + 5666*B*b^3*m*x^9*x^m + 16998*A*b^2*c*m*x^9*x^m + 854*A*b^3*m^2*
x^7*x^m + 15015*B*b^3*x^9*x^m + 45045*A*b^2*c*x^9*x^m + 6672*A*b^3*m*x^7*x^m + 19305*A*b^3*x^7*x^m)/(m^5 + 55*
m^4 + 1190*m^3 + 12650*m^2 + 66009*m + 135135)