Integrand size = 27, antiderivative size = 179 \[ \int x^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {a^6 A x^{1+m}}{1+m}+\frac {a^5 (6 A b+a B) x^{2+m}}{2+m}+\frac {3 a^4 b (5 A b+2 a B) x^{3+m}}{3+m}+\frac {5 a^3 b^2 (4 A b+3 a B) x^{4+m}}{4+m}+\frac {5 a^2 b^3 (3 A b+4 a B) x^{5+m}}{5+m}+\frac {3 a b^4 (2 A b+5 a B) x^{6+m}}{6+m}+\frac {b^5 (A b+6 a B) x^{7+m}}{7+m}+\frac {b^6 B x^{8+m}}{8+m} \]
a^6*A*x^(1+m)/(1+m)+a^5*(6*A*b+B*a)*x^(2+m)/(2+m)+3*a^4*b*(5*A*b+2*B*a)*x^ (3+m)/(3+m)+5*a^3*b^2*(4*A*b+3*B*a)*x^(4+m)/(4+m)+5*a^2*b^3*(3*A*b+4*B*a)* x^(5+m)/(5+m)+3*a*b^4*(2*A*b+5*B*a)*x^(6+m)/(6+m)+b^5*(A*b+6*B*a)*x^(7+m)/ (7+m)+b^6*B*x^(8+m)/(8+m)
Time = 0.36 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.75 \[ \int x^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {x^{1+m} \left (B (a+b x)^7+(-a B (1+m)+A b (8+m)) \left (\frac {a^6}{1+m}+\frac {6 a^5 b x}{2+m}+\frac {15 a^4 b^2 x^2}{3+m}+\frac {20 a^3 b^3 x^3}{4+m}+\frac {15 a^2 b^4 x^4}{5+m}+\frac {6 a b^5 x^5}{6+m}+\frac {b^6 x^6}{7+m}\right )\right )}{b (8+m)} \]
(x^(1 + m)*(B*(a + b*x)^7 + (-(a*B*(1 + m)) + A*b*(8 + m))*(a^6/(1 + m) + (6*a^5*b*x)/(2 + m) + (15*a^4*b^2*x^2)/(3 + m) + (20*a^3*b^3*x^3)/(4 + m) + (15*a^2*b^4*x^4)/(5 + m) + (6*a*b^5*x^5)/(6 + m) + (b^6*x^6)/(7 + m))))/ (b*(8 + m))
Time = 0.36 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1184, 27, 85, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^m \left (a^2+2 a b x+b^2 x^2\right )^3 (A+B x) \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \frac {\int b^6 x^m (a+b x)^6 (A+B x)dx}{b^6}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int x^m (a+b x)^6 (A+B x)dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left (a^6 A x^m+a^5 x^{m+1} (a B+6 A b)+3 a^4 b x^{m+2} (2 a B+5 A b)+5 a^3 b^2 x^{m+3} (3 a B+4 A b)+5 a^2 b^3 x^{m+4} (4 a B+3 A b)+b^5 x^{m+6} (6 a B+A b)+3 a b^4 x^{m+5} (5 a B+2 A b)+b^6 B x^{m+7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^6 A x^{m+1}}{m+1}+\frac {a^5 x^{m+2} (a B+6 A b)}{m+2}+\frac {3 a^4 b x^{m+3} (2 a B+5 A b)}{m+3}+\frac {5 a^3 b^2 x^{m+4} (3 a B+4 A b)}{m+4}+\frac {5 a^2 b^3 x^{m+5} (4 a B+3 A b)}{m+5}+\frac {b^5 x^{m+7} (6 a B+A b)}{m+7}+\frac {3 a b^4 x^{m+6} (5 a B+2 A b)}{m+6}+\frac {b^6 B x^{m+8}}{m+8}\) |
(a^6*A*x^(1 + m))/(1 + m) + (a^5*(6*A*b + a*B)*x^(2 + m))/(2 + m) + (3*a^4 *b*(5*A*b + 2*a*B)*x^(3 + m))/(3 + m) + (5*a^3*b^2*(4*A*b + 3*a*B)*x^(4 + m))/(4 + m) + (5*a^2*b^3*(3*A*b + 4*a*B)*x^(5 + m))/(5 + m) + (3*a*b^4*(2* A*b + 5*a*B)*x^(6 + m))/(6 + m) + (b^5*(A*b + 6*a*B)*x^(7 + m))/(7 + m) + (b^6*B*x^(8 + m))/(8 + m)
3.9.36.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.33 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.13
method | result | size |
norman | \(\frac {A \,a^{6} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {B \,b^{6} x^{8} {\mathrm e}^{m \ln \left (x \right )}}{8+m}+\frac {a^{5} \left (6 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {b^{5} \left (A b +6 B a \right ) x^{7} {\mathrm e}^{m \ln \left (x \right )}}{7+m}+\frac {5 a^{2} b^{3} \left (3 A b +4 B a \right ) x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {5 a^{3} b^{2} \left (4 A b +3 B a \right ) x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {3 a^{4} b \left (5 A b +2 B a \right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {3 b^{4} a \left (2 A b +5 B a \right ) x^{6} {\mathrm e}^{m \ln \left (x \right )}}{6+m}\) | \(202\) |
risch | \(\text {Expression too large to display}\) | \(1437\) |
gosper | \(\text {Expression too large to display}\) | \(1438\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1809\) |
A*a^6/(1+m)*x*exp(m*ln(x))+B*b^6/(8+m)*x^8*exp(m*ln(x))+a^5*(6*A*b+B*a)/(2 +m)*x^2*exp(m*ln(x))+b^5*(A*b+6*B*a)/(7+m)*x^7*exp(m*ln(x))+5*a^2*b^3*(3*A *b+4*B*a)/(5+m)*x^5*exp(m*ln(x))+5*a^3*b^2*(4*A*b+3*B*a)/(4+m)*x^4*exp(m*l n(x))+3*a^4*b*(5*A*b+2*B*a)/(3+m)*x^3*exp(m*ln(x))+3*b^4*a*(2*A*b+5*B*a)/( 6+m)*x^6*exp(m*ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 1191 vs. \(2 (179) = 358\).
Time = 0.32 (sec) , antiderivative size = 1191, normalized size of antiderivative = 6.65 \[ \int x^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
((B*b^6*m^7 + 28*B*b^6*m^6 + 322*B*b^6*m^5 + 1960*B*b^6*m^4 + 6769*B*b^6*m ^3 + 13132*B*b^6*m^2 + 13068*B*b^6*m + 5040*B*b^6)*x^8 + ((6*B*a*b^5 + A*b ^6)*m^7 + 34560*B*a*b^5 + 5760*A*b^6 + 29*(6*B*a*b^5 + A*b^6)*m^6 + 343*(6 *B*a*b^5 + A*b^6)*m^5 + 2135*(6*B*a*b^5 + A*b^6)*m^4 + 7504*(6*B*a*b^5 + A *b^6)*m^3 + 14756*(6*B*a*b^5 + A*b^6)*m^2 + 14832*(6*B*a*b^5 + A*b^6)*m)*x ^7 + 3*((5*B*a^2*b^4 + 2*A*a*b^5)*m^7 + 33600*B*a^2*b^4 + 13440*A*a*b^5 + 30*(5*B*a^2*b^4 + 2*A*a*b^5)*m^6 + 366*(5*B*a^2*b^4 + 2*A*a*b^5)*m^5 + 234 0*(5*B*a^2*b^4 + 2*A*a*b^5)*m^4 + 8409*(5*B*a^2*b^4 + 2*A*a*b^5)*m^3 + 168 30*(5*B*a^2*b^4 + 2*A*a*b^5)*m^2 + 17144*(5*B*a^2*b^4 + 2*A*a*b^5)*m)*x^6 + 5*((4*B*a^3*b^3 + 3*A*a^2*b^4)*m^7 + 32256*B*a^3*b^3 + 24192*A*a^2*b^4 + 31*(4*B*a^3*b^3 + 3*A*a^2*b^4)*m^6 + 391*(4*B*a^3*b^3 + 3*A*a^2*b^4)*m^5 + 2581*(4*B*a^3*b^3 + 3*A*a^2*b^4)*m^4 + 9544*(4*B*a^3*b^3 + 3*A*a^2*b^4)* m^3 + 19564*(4*B*a^3*b^3 + 3*A*a^2*b^4)*m^2 + 20304*(4*B*a^3*b^3 + 3*A*a^2 *b^4)*m)*x^5 + 5*((3*B*a^4*b^2 + 4*A*a^3*b^3)*m^7 + 30240*B*a^4*b^2 + 4032 0*A*a^3*b^3 + 32*(3*B*a^4*b^2 + 4*A*a^3*b^3)*m^6 + 418*(3*B*a^4*b^2 + 4*A* a^3*b^3)*m^5 + 2864*(3*B*a^4*b^2 + 4*A*a^3*b^3)*m^4 + 10993*(3*B*a^4*b^2 + 4*A*a^3*b^3)*m^3 + 23312*(3*B*a^4*b^2 + 4*A*a^3*b^3)*m^2 + 24876*(3*B*a^4 *b^2 + 4*A*a^3*b^3)*m)*x^4 + 3*((2*B*a^5*b + 5*A*a^4*b^2)*m^7 + 26880*B*a^ 5*b + 67200*A*a^4*b^2 + 33*(2*B*a^5*b + 5*A*a^4*b^2)*m^6 + 447*(2*B*a^5*b + 5*A*a^4*b^2)*m^5 + 3195*(2*B*a^5*b + 5*A*a^4*b^2)*m^4 + 12864*(2*B*a^...
Leaf count of result is larger than twice the leaf count of optimal. 7745 vs. \(2 (172) = 344\).
Time = 0.84 (sec) , antiderivative size = 7745, normalized size of antiderivative = 43.27 \[ \int x^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise((-A*a**6/(7*x**7) - A*a**5*b/x**6 - 3*A*a**4*b**2/x**5 - 5*A*a** 3*b**3/x**4 - 5*A*a**2*b**4/x**3 - 3*A*a*b**5/x**2 - A*b**6/x - B*a**6/(6* x**6) - 6*B*a**5*b/(5*x**5) - 15*B*a**4*b**2/(4*x**4) - 20*B*a**3*b**3/(3* x**3) - 15*B*a**2*b**4/(2*x**2) - 6*B*a*b**5/x + B*b**6*log(x), Eq(m, -8)) , (-A*a**6/(6*x**6) - 6*A*a**5*b/(5*x**5) - 15*A*a**4*b**2/(4*x**4) - 20*A *a**3*b**3/(3*x**3) - 15*A*a**2*b**4/(2*x**2) - 6*A*a*b**5/x + A*b**6*log( x) - B*a**6/(5*x**5) - 3*B*a**5*b/(2*x**4) - 5*B*a**4*b**2/x**3 - 10*B*a** 3*b**3/x**2 - 15*B*a**2*b**4/x + 6*B*a*b**5*log(x) + B*b**6*x, Eq(m, -7)), (-A*a**6/(5*x**5) - 3*A*a**5*b/(2*x**4) - 5*A*a**4*b**2/x**3 - 10*A*a**3* b**3/x**2 - 15*A*a**2*b**4/x + 6*A*a*b**5*log(x) + A*b**6*x - B*a**6/(4*x* *4) - 2*B*a**5*b/x**3 - 15*B*a**4*b**2/(2*x**2) - 20*B*a**3*b**3/x + 15*B* a**2*b**4*log(x) + 6*B*a*b**5*x + B*b**6*x**2/2, Eq(m, -6)), (-A*a**6/(4*x **4) - 2*A*a**5*b/x**3 - 15*A*a**4*b**2/(2*x**2) - 20*A*a**3*b**3/x + 15*A *a**2*b**4*log(x) + 6*A*a*b**5*x + A*b**6*x**2/2 - B*a**6/(3*x**3) - 3*B*a **5*b/x**2 - 15*B*a**4*b**2/x + 20*B*a**3*b**3*log(x) + 15*B*a**2*b**4*x + 3*B*a*b**5*x**2 + B*b**6*x**3/3, Eq(m, -5)), (-A*a**6/(3*x**3) - 3*A*a**5 *b/x**2 - 15*A*a**4*b**2/x + 20*A*a**3*b**3*log(x) + 15*A*a**2*b**4*x + 3* A*a*b**5*x**2 + A*b**6*x**3/3 - B*a**6/(2*x**2) - 6*B*a**5*b/x + 15*B*a**4 *b**2*log(x) + 20*B*a**3*b**3*x + 15*B*a**2*b**4*x**2/2 + 2*B*a*b**5*x**3 + B*b**6*x**4/4, Eq(m, -4)), (-A*a**6/(2*x**2) - 6*A*a**5*b/x + 15*A*a*...
Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.36 \[ \int x^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {B b^{6} x^{m + 8}}{m + 8} + \frac {6 \, B a b^{5} x^{m + 7}}{m + 7} + \frac {A b^{6} x^{m + 7}}{m + 7} + \frac {15 \, B a^{2} b^{4} x^{m + 6}}{m + 6} + \frac {6 \, A a b^{5} x^{m + 6}}{m + 6} + \frac {20 \, B a^{3} b^{3} x^{m + 5}}{m + 5} + \frac {15 \, A a^{2} b^{4} x^{m + 5}}{m + 5} + \frac {15 \, B a^{4} b^{2} x^{m + 4}}{m + 4} + \frac {20 \, A a^{3} b^{3} x^{m + 4}}{m + 4} + \frac {6 \, B a^{5} b x^{m + 3}}{m + 3} + \frac {15 \, A a^{4} b^{2} x^{m + 3}}{m + 3} + \frac {B a^{6} x^{m + 2}}{m + 2} + \frac {6 \, A a^{5} b x^{m + 2}}{m + 2} + \frac {A a^{6} x^{m + 1}}{m + 1} \]
B*b^6*x^(m + 8)/(m + 8) + 6*B*a*b^5*x^(m + 7)/(m + 7) + A*b^6*x^(m + 7)/(m + 7) + 15*B*a^2*b^4*x^(m + 6)/(m + 6) + 6*A*a*b^5*x^(m + 6)/(m + 6) + 20* B*a^3*b^3*x^(m + 5)/(m + 5) + 15*A*a^2*b^4*x^(m + 5)/(m + 5) + 15*B*a^4*b^ 2*x^(m + 4)/(m + 4) + 20*A*a^3*b^3*x^(m + 4)/(m + 4) + 6*B*a^5*b*x^(m + 3) /(m + 3) + 15*A*a^4*b^2*x^(m + 3)/(m + 3) + B*a^6*x^(m + 2)/(m + 2) + 6*A* a^5*b*x^(m + 2)/(m + 2) + A*a^6*x^(m + 1)/(m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1808 vs. \(2 (179) = 358\).
Time = 0.29 (sec) , antiderivative size = 1808, normalized size of antiderivative = 10.10 \[ \int x^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\text {Too large to display} \]
(B*b^6*m^7*x^8*x^m + 6*B*a*b^5*m^7*x^7*x^m + A*b^6*m^7*x^7*x^m + 28*B*b^6* m^6*x^8*x^m + 15*B*a^2*b^4*m^7*x^6*x^m + 6*A*a*b^5*m^7*x^6*x^m + 174*B*a*b ^5*m^6*x^7*x^m + 29*A*b^6*m^6*x^7*x^m + 322*B*b^6*m^5*x^8*x^m + 20*B*a^3*b ^3*m^7*x^5*x^m + 15*A*a^2*b^4*m^7*x^5*x^m + 450*B*a^2*b^4*m^6*x^6*x^m + 18 0*A*a*b^5*m^6*x^6*x^m + 2058*B*a*b^5*m^5*x^7*x^m + 343*A*b^6*m^5*x^7*x^m + 1960*B*b^6*m^4*x^8*x^m + 15*B*a^4*b^2*m^7*x^4*x^m + 20*A*a^3*b^3*m^7*x^4* x^m + 620*B*a^3*b^3*m^6*x^5*x^m + 465*A*a^2*b^4*m^6*x^5*x^m + 5490*B*a^2*b ^4*m^5*x^6*x^m + 2196*A*a*b^5*m^5*x^6*x^m + 12810*B*a*b^5*m^4*x^7*x^m + 21 35*A*b^6*m^4*x^7*x^m + 6769*B*b^6*m^3*x^8*x^m + 6*B*a^5*b*m^7*x^3*x^m + 15 *A*a^4*b^2*m^7*x^3*x^m + 480*B*a^4*b^2*m^6*x^4*x^m + 640*A*a^3*b^3*m^6*x^4 *x^m + 7820*B*a^3*b^3*m^5*x^5*x^m + 5865*A*a^2*b^4*m^5*x^5*x^m + 35100*B*a ^2*b^4*m^4*x^6*x^m + 14040*A*a*b^5*m^4*x^6*x^m + 45024*B*a*b^5*m^3*x^7*x^m + 7504*A*b^6*m^3*x^7*x^m + 13132*B*b^6*m^2*x^8*x^m + B*a^6*m^7*x^2*x^m + 6*A*a^5*b*m^7*x^2*x^m + 198*B*a^5*b*m^6*x^3*x^m + 495*A*a^4*b^2*m^6*x^3*x^ m + 6270*B*a^4*b^2*m^5*x^4*x^m + 8360*A*a^3*b^3*m^5*x^4*x^m + 51620*B*a^3* b^3*m^4*x^5*x^m + 38715*A*a^2*b^4*m^4*x^5*x^m + 126135*B*a^2*b^4*m^3*x^6*x ^m + 50454*A*a*b^5*m^3*x^6*x^m + 88536*B*a*b^5*m^2*x^7*x^m + 14756*A*b^6*m ^2*x^7*x^m + 13068*B*b^6*m*x^8*x^m + A*a^6*m^7*x*x^m + 34*B*a^6*m^6*x^2*x^ m + 204*A*a^5*b*m^6*x^2*x^m + 2682*B*a^5*b*m^5*x^3*x^m + 6705*A*a^4*b^2*m^ 5*x^3*x^m + 42960*B*a^4*b^2*m^4*x^4*x^m + 57280*A*a^3*b^3*m^4*x^4*x^m +...
Time = 10.68 (sec) , antiderivative size = 729, normalized size of antiderivative = 4.07 \[ \int x^m (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {A\,a^6\,x\,x^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {B\,b^6\,x^m\,x^8\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a^5\,x^m\,x^2\,\left (6\,A\,b+B\,a\right )\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {b^5\,x^m\,x^7\,\left (A\,b+6\,B\,a\right )\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,a\,b^4\,x^m\,x^6\,\left (2\,A\,b+5\,B\,a\right )\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,a^4\,b\,x^m\,x^3\,\left (5\,A\,b+2\,B\,a\right )\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {5\,a^2\,b^3\,x^m\,x^5\,\left (3\,A\,b+4\,B\,a\right )\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {5\,a^3\,b^2\,x^m\,x^4\,\left (4\,A\,b+3\,B\,a\right )\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \]
(A*a^6*x*x^m*(69264*m + 48860*m^2 + 18424*m^3 + 4025*m^4 + 511*m^5 + 35*m^ 6 + m^7 + 40320))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^ 5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (B*b^6*x^m*x^8*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040))/(109584*m + 118124 *m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (a^5*x^m*x^2*(6*A*b + B*a)*(44712*m + 36706*m^2 + 15289*m^3 + 3580*m^4 + 478*m^5 + 34*m^6 + m^7 + 20160))/(109584*m + 118124*m^2 + 67284*m^3 + 224 49*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (b^5*x^m*x^7*(A*b + 6*B*a)*(14832*m + 14756*m^2 + 7504*m^3 + 2135*m^4 + 343*m^5 + 29*m^6 + m^7 + 5760))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546* m^6 + 36*m^7 + m^8 + 40320) + (3*a*b^4*x^m*x^6*(2*A*b + 5*B*a)*(17144*m + 16830*m^2 + 8409*m^3 + 2340*m^4 + 366*m^5 + 30*m^6 + m^7 + 6720))/(109584* m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (3*a^4*b*x^m*x^3*(5*A*b + 2*B*a)*(32048*m + 28692*m^2 + 12864* m^3 + 3195*m^4 + 447*m^5 + 33*m^6 + m^7 + 13440))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (5*a ^2*b^3*x^m*x^5*(3*A*b + 4*B*a)*(20304*m + 19564*m^2 + 9544*m^3 + 2581*m^4 + 391*m^5 + 31*m^6 + m^7 + 8064))/(109584*m + 118124*m^2 + 67284*m^3 + 224 49*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (5*a^3*b^2*x^m*x^4*( 4*A*b + 3*B*a)*(24876*m + 23312*m^2 + 10993*m^3 + 2864*m^4 + 418*m^5 + ...