Integrand size = 18, antiderivative size = 231 \[ \int x^m (a+b x)^2 (c+d x)^5 \, dx=\frac {a^2 c^5 x^{1+m}}{1+m}+\frac {a c^4 (2 b c+5 a d) x^{2+m}}{2+m}+\frac {c^3 \left (b^2 c^2+10 a b c d+10 a^2 d^2\right ) x^{3+m}}{3+m}+\frac {5 c^2 d \left (b^2 c^2+4 a b c d+2 a^2 d^2\right ) x^{4+m}}{4+m}+\frac {5 c d^2 \left (2 b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5+m}}{5+m}+\frac {d^3 \left (10 b^2 c^2+10 a b c d+a^2 d^2\right ) x^{6+m}}{6+m}+\frac {b d^4 (5 b c+2 a d) x^{7+m}}{7+m}+\frac {b^2 d^5 x^{8+m}}{8+m} \]
a^2*c^5*x^(1+m)/(1+m)+a*c^4*(5*a*d+2*b*c)*x^(2+m)/(2+m)+c^3*(10*a^2*d^2+10 *a*b*c*d+b^2*c^2)*x^(3+m)/(3+m)+5*c^2*d*(2*a^2*d^2+4*a*b*c*d+b^2*c^2)*x^(4 +m)/(4+m)+5*c*d^2*(a^2*d^2+4*a*b*c*d+2*b^2*c^2)*x^(5+m)/(5+m)+d^3*(a^2*d^2 +10*a*b*c*d+10*b^2*c^2)*x^(6+m)/(6+m)+b*d^4*(2*a*d+5*b*c)*x^(7+m)/(7+m)+b^ 2*d^5*x^(8+m)/(8+m)
Time = 0.47 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.94 \[ \int x^m (a+b x)^2 (c+d x)^5 \, dx=x^{1+m} \left (\frac {a^2 c^5}{1+m}+\frac {a c^4 (2 b c+5 a d) x}{2+m}+\frac {c^3 \left (b^2 c^2+10 a b c d+10 a^2 d^2\right ) x^2}{3+m}+\frac {5 c^2 d \left (b^2 c^2+4 a b c d+2 a^2 d^2\right ) x^3}{4+m}+\frac {5 c d^2 \left (2 b^2 c^2+4 a b c d+a^2 d^2\right ) x^4}{5+m}+\frac {d^3 \left (10 b^2 c^2+10 a b c d+a^2 d^2\right ) x^5}{6+m}+\frac {b d^4 (5 b c+2 a d) x^6}{7+m}+\frac {b^2 d^5 x^7}{8+m}\right ) \]
x^(1 + m)*((a^2*c^5)/(1 + m) + (a*c^4*(2*b*c + 5*a*d)*x)/(2 + m) + (c^3*(b ^2*c^2 + 10*a*b*c*d + 10*a^2*d^2)*x^2)/(3 + m) + (5*c^2*d*(b^2*c^2 + 4*a*b *c*d + 2*a^2*d^2)*x^3)/(4 + m) + (5*c*d^2*(2*b^2*c^2 + 4*a*b*c*d + a^2*d^2 )*x^4)/(5 + m) + (d^3*(10*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*x^5)/(6 + m) + ( b*d^4*(5*b*c + 2*a*d)*x^6)/(7 + m) + (b^2*d^5*x^7)/(8 + m))
Time = 0.38 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 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.
| \(\displaystyle \int x^m (a+b x)^2 (c+d x)^5 \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (5 c^2 d x^{m+3} \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )+5 c d^2 x^{m+4} \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )+d^3 x^{m+5} \left (a^2 d^2+10 a b c d+10 b^2 c^2\right )+c^3 x^{m+2} \left (10 a^2 d^2+10 a b c d+b^2 c^2\right )+a^2 c^5 x^m+a c^4 x^{m+1} (5 a d+2 b c)+b d^4 x^{m+6} (2 a d+5 b c)+b^2 d^5 x^{m+7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 c^2 d x^{m+4} \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )}{m+4}+\frac {5 c d^2 x^{m+5} \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )}{m+5}+\frac {d^3 x^{m+6} \left (a^2 d^2+10 a b c d+10 b^2 c^2\right )}{m+6}+\frac {c^3 x^{m+3} \left (10 a^2 d^2+10 a b c d+b^2 c^2\right )}{m+3}+\frac {a^2 c^5 x^{m+1}}{m+1}+\frac {a c^4 x^{m+2} (5 a d+2 b c)}{m+2}+\frac {b d^4 x^{m+7} (2 a d+5 b c)}{m+7}+\frac {b^2 d^5 x^{m+8}}{m+8}\) |
(a^2*c^5*x^(1 + m))/(1 + m) + (a*c^4*(2*b*c + 5*a*d)*x^(2 + m))/(2 + m) + (c^3*(b^2*c^2 + 10*a*b*c*d + 10*a^2*d^2)*x^(3 + m))/(3 + m) + (5*c^2*d*(b^ 2*c^2 + 4*a*b*c*d + 2*a^2*d^2)*x^(4 + m))/(4 + m) + (5*c*d^2*(2*b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5 + m))/(5 + m) + (d^3*(10*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*x^(6 + m))/(6 + m) + (b*d^4*(5*b*c + 2*a*d)*x^(7 + m))/(7 + m) + (b^2*d^5*x^(8 + m))/(8 + m)
3.4.78.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.56 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.10
| method | result | size |
| norman | \(\frac {a^{2} c^{5} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{2} d^{5} x^{8} {\mathrm e}^{m \ln \left (x \right )}}{8+m}+\frac {c^{3} \left (10 a^{2} d^{2}+10 a b c d +b^{2} c^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {d^{3} \left (a^{2} d^{2}+10 a b c d +10 b^{2} c^{2}\right ) x^{6} {\mathrm e}^{m \ln \left (x \right )}}{6+m}+\frac {a \,c^{4} \left (5 a d +2 b c \right ) x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {b \,d^{4} \left (2 a d +5 b c \right ) x^{7} {\mathrm e}^{m \ln \left (x \right )}}{7+m}+\frac {5 c \,d^{2} \left (a^{2} d^{2}+4 a b c d +2 b^{2} c^{2}\right ) x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {5 c^{2} d \left (2 a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}\) | \(254\) |
| risch | \(\text {Expression too large to display}\) | \(2057\) |
| gosper | \(\text {Expression too large to display}\) | \(2058\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2525\) |
a^2*c^5/(1+m)*x*exp(m*ln(x))+b^2*d^5/(8+m)*x^8*exp(m*ln(x))+c^3*(10*a^2*d^ 2+10*a*b*c*d+b^2*c^2)/(3+m)*x^3*exp(m*ln(x))+d^3*(a^2*d^2+10*a*b*c*d+10*b^ 2*c^2)/(6+m)*x^6*exp(m*ln(x))+a*c^4*(5*a*d+2*b*c)/(2+m)*x^2*exp(m*ln(x))+b *d^4*(2*a*d+5*b*c)/(7+m)*x^7*exp(m*ln(x))+5*c*d^2*(a^2*d^2+4*a*b*c*d+2*b^2 *c^2)/(5+m)*x^5*exp(m*ln(x))+5*c^2*d*(2*a^2*d^2+4*a*b*c*d+b^2*c^2)/(4+m)*x ^4*exp(m*ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 1623 vs. \(2 (231) = 462\).
Time = 0.26 (sec) , antiderivative size = 1623, normalized size of antiderivative = 7.03 \[ \int x^m (a+b x)^2 (c+d x)^5 \, dx=\text {Too large to display} \]
((b^2*d^5*m^7 + 28*b^2*d^5*m^6 + 322*b^2*d^5*m^5 + 1960*b^2*d^5*m^4 + 6769 *b^2*d^5*m^3 + 13132*b^2*d^5*m^2 + 13068*b^2*d^5*m + 5040*b^2*d^5)*x^8 + ( (5*b^2*c*d^4 + 2*a*b*d^5)*m^7 + 28800*b^2*c*d^4 + 11520*a*b*d^5 + 29*(5*b^ 2*c*d^4 + 2*a*b*d^5)*m^6 + 343*(5*b^2*c*d^4 + 2*a*b*d^5)*m^5 + 2135*(5*b^2 *c*d^4 + 2*a*b*d^5)*m^4 + 7504*(5*b^2*c*d^4 + 2*a*b*d^5)*m^3 + 14756*(5*b^ 2*c*d^4 + 2*a*b*d^5)*m^2 + 14832*(5*b^2*c*d^4 + 2*a*b*d^5)*m)*x^7 + ((10*b ^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m^7 + 67200*b^2*c^2*d^3 + 67200*a*b*c *d^4 + 6720*a^2*d^5 + 30*(10*b^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m^6 + 3 66*(10*b^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m^5 + 2340*(10*b^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m^4 + 8409*(10*b^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^ 5)*m^3 + 16830*(10*b^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m^2 + 17144*(10*b ^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m)*x^6 + 5*((2*b^2*c^3*d^2 + 4*a*b*c^ 2*d^3 + a^2*c*d^4)*m^7 + 16128*b^2*c^3*d^2 + 32256*a*b*c^2*d^3 + 8064*a^2* c*d^4 + 31*(2*b^2*c^3*d^2 + 4*a*b*c^2*d^3 + a^2*c*d^4)*m^6 + 391*(2*b^2*c^ 3*d^2 + 4*a*b*c^2*d^3 + a^2*c*d^4)*m^5 + 2581*(2*b^2*c^3*d^2 + 4*a*b*c^2*d ^3 + a^2*c*d^4)*m^4 + 9544*(2*b^2*c^3*d^2 + 4*a*b*c^2*d^3 + a^2*c*d^4)*m^3 + 19564*(2*b^2*c^3*d^2 + 4*a*b*c^2*d^3 + a^2*c*d^4)*m^2 + 20304*(2*b^2*c^ 3*d^2 + 4*a*b*c^2*d^3 + a^2*c*d^4)*m)*x^5 + 5*((b^2*c^4*d + 4*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*m^7 + 10080*b^2*c^4*d + 40320*a*b*c^3*d^2 + 20160*a^2*c^2 *d^3 + 32*(b^2*c^4*d + 4*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*m^6 + 418*(b^2*c^...
Leaf count of result is larger than twice the leaf count of optimal. 10401 vs. \(2 (226) = 452\).
Time = 0.98 (sec) , antiderivative size = 10401, normalized size of antiderivative = 45.03 \[ \int x^m (a+b x)^2 (c+d x)^5 \, dx=\text {Too large to display} \]
Piecewise((-a**2*c**5/(7*x**7) - 5*a**2*c**4*d/(6*x**6) - 2*a**2*c**3*d**2 /x**5 - 5*a**2*c**2*d**3/(2*x**4) - 5*a**2*c*d**4/(3*x**3) - a**2*d**5/(2* x**2) - a*b*c**5/(3*x**6) - 2*a*b*c**4*d/x**5 - 5*a*b*c**3*d**2/x**4 - 20* a*b*c**2*d**3/(3*x**3) - 5*a*b*c*d**4/x**2 - 2*a*b*d**5/x - b**2*c**5/(5*x **5) - 5*b**2*c**4*d/(4*x**4) - 10*b**2*c**3*d**2/(3*x**3) - 5*b**2*c**2*d **3/x**2 - 5*b**2*c*d**4/x + b**2*d**5*log(x), Eq(m, -8)), (-a**2*c**5/(6* x**6) - a**2*c**4*d/x**5 - 5*a**2*c**3*d**2/(2*x**4) - 10*a**2*c**2*d**3/( 3*x**3) - 5*a**2*c*d**4/(2*x**2) - a**2*d**5/x - 2*a*b*c**5/(5*x**5) - 5*a *b*c**4*d/(2*x**4) - 20*a*b*c**3*d**2/(3*x**3) - 10*a*b*c**2*d**3/x**2 - 1 0*a*b*c*d**4/x + 2*a*b*d**5*log(x) - b**2*c**5/(4*x**4) - 5*b**2*c**4*d/(3 *x**3) - 5*b**2*c**3*d**2/x**2 - 10*b**2*c**2*d**3/x + 5*b**2*c*d**4*log(x ) + b**2*d**5*x, Eq(m, -7)), (-a**2*c**5/(5*x**5) - 5*a**2*c**4*d/(4*x**4) - 10*a**2*c**3*d**2/(3*x**3) - 5*a**2*c**2*d**3/x**2 - 5*a**2*c*d**4/x + a**2*d**5*log(x) - a*b*c**5/(2*x**4) - 10*a*b*c**4*d/(3*x**3) - 10*a*b*c** 3*d**2/x**2 - 20*a*b*c**2*d**3/x + 10*a*b*c*d**4*log(x) + 2*a*b*d**5*x - b **2*c**5/(3*x**3) - 5*b**2*c**4*d/(2*x**2) - 10*b**2*c**3*d**2/x + 10*b**2 *c**2*d**3*log(x) + 5*b**2*c*d**4*x + b**2*d**5*x**2/2, Eq(m, -6)), (-a**2 *c**5/(4*x**4) - 5*a**2*c**4*d/(3*x**3) - 5*a**2*c**3*d**2/x**2 - 10*a**2* c**2*d**3/x + 5*a**2*c*d**4*log(x) + a**2*d**5*x - 2*a*b*c**5/(3*x**3) - 5 *a*b*c**4*d/x**2 - 20*a*b*c**3*d**2/x + 20*a*b*c**2*d**3*log(x) + 10*a*...
Time = 0.20 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.47 \[ \int x^m (a+b x)^2 (c+d x)^5 \, dx=\frac {b^{2} d^{5} x^{m + 8}}{m + 8} + \frac {5 \, b^{2} c d^{4} x^{m + 7}}{m + 7} + \frac {2 \, a b d^{5} x^{m + 7}}{m + 7} + \frac {10 \, b^{2} c^{2} d^{3} x^{m + 6}}{m + 6} + \frac {10 \, a b c d^{4} x^{m + 6}}{m + 6} + \frac {a^{2} d^{5} x^{m + 6}}{m + 6} + \frac {10 \, b^{2} c^{3} d^{2} x^{m + 5}}{m + 5} + \frac {20 \, a b c^{2} d^{3} x^{m + 5}}{m + 5} + \frac {5 \, a^{2} c d^{4} x^{m + 5}}{m + 5} + \frac {5 \, b^{2} c^{4} d x^{m + 4}}{m + 4} + \frac {20 \, a b c^{3} d^{2} x^{m + 4}}{m + 4} + \frac {10 \, a^{2} c^{2} d^{3} x^{m + 4}}{m + 4} + \frac {b^{2} c^{5} x^{m + 3}}{m + 3} + \frac {10 \, a b c^{4} d x^{m + 3}}{m + 3} + \frac {10 \, a^{2} c^{3} d^{2} x^{m + 3}}{m + 3} + \frac {2 \, a b c^{5} x^{m + 2}}{m + 2} + \frac {5 \, a^{2} c^{4} d x^{m + 2}}{m + 2} + \frac {a^{2} c^{5} x^{m + 1}}{m + 1} \]
b^2*d^5*x^(m + 8)/(m + 8) + 5*b^2*c*d^4*x^(m + 7)/(m + 7) + 2*a*b*d^5*x^(m + 7)/(m + 7) + 10*b^2*c^2*d^3*x^(m + 6)/(m + 6) + 10*a*b*c*d^4*x^(m + 6)/ (m + 6) + a^2*d^5*x^(m + 6)/(m + 6) + 10*b^2*c^3*d^2*x^(m + 5)/(m + 5) + 2 0*a*b*c^2*d^3*x^(m + 5)/(m + 5) + 5*a^2*c*d^4*x^(m + 5)/(m + 5) + 5*b^2*c^ 4*d*x^(m + 4)/(m + 4) + 20*a*b*c^3*d^2*x^(m + 4)/(m + 4) + 10*a^2*c^2*d^3* x^(m + 4)/(m + 4) + b^2*c^5*x^(m + 3)/(m + 3) + 10*a*b*c^4*d*x^(m + 3)/(m + 3) + 10*a^2*c^3*d^2*x^(m + 3)/(m + 3) + 2*a*b*c^5*x^(m + 2)/(m + 2) + 5* a^2*c^4*d*x^(m + 2)/(m + 2) + a^2*c^5*x^(m + 1)/(m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 2524 vs. \(2 (231) = 462\).
Time = 0.31 (sec) , antiderivative size = 2524, normalized size of antiderivative = 10.93 \[ \int x^m (a+b x)^2 (c+d x)^5 \, dx=\text {Too large to display} \]
(b^2*d^5*m^7*x^8*x^m + 5*b^2*c*d^4*m^7*x^7*x^m + 2*a*b*d^5*m^7*x^7*x^m + 2 8*b^2*d^5*m^6*x^8*x^m + 10*b^2*c^2*d^3*m^7*x^6*x^m + 10*a*b*c*d^4*m^7*x^6* x^m + a^2*d^5*m^7*x^6*x^m + 145*b^2*c*d^4*m^6*x^7*x^m + 58*a*b*d^5*m^6*x^7 *x^m + 322*b^2*d^5*m^5*x^8*x^m + 10*b^2*c^3*d^2*m^7*x^5*x^m + 20*a*b*c^2*d ^3*m^7*x^5*x^m + 5*a^2*c*d^4*m^7*x^5*x^m + 300*b^2*c^2*d^3*m^6*x^6*x^m + 3 00*a*b*c*d^4*m^6*x^6*x^m + 30*a^2*d^5*m^6*x^6*x^m + 1715*b^2*c*d^4*m^5*x^7 *x^m + 686*a*b*d^5*m^5*x^7*x^m + 1960*b^2*d^5*m^4*x^8*x^m + 5*b^2*c^4*d*m^ 7*x^4*x^m + 20*a*b*c^3*d^2*m^7*x^4*x^m + 10*a^2*c^2*d^3*m^7*x^4*x^m + 310* b^2*c^3*d^2*m^6*x^5*x^m + 620*a*b*c^2*d^3*m^6*x^5*x^m + 155*a^2*c*d^4*m^6* x^5*x^m + 3660*b^2*c^2*d^3*m^5*x^6*x^m + 3660*a*b*c*d^4*m^5*x^6*x^m + 366* a^2*d^5*m^5*x^6*x^m + 10675*b^2*c*d^4*m^4*x^7*x^m + 4270*a*b*d^5*m^4*x^7*x ^m + 6769*b^2*d^5*m^3*x^8*x^m + b^2*c^5*m^7*x^3*x^m + 10*a*b*c^4*d*m^7*x^3 *x^m + 10*a^2*c^3*d^2*m^7*x^3*x^m + 160*b^2*c^4*d*m^6*x^4*x^m + 640*a*b*c^ 3*d^2*m^6*x^4*x^m + 320*a^2*c^2*d^3*m^6*x^4*x^m + 3910*b^2*c^3*d^2*m^5*x^5 *x^m + 7820*a*b*c^2*d^3*m^5*x^5*x^m + 1955*a^2*c*d^4*m^5*x^5*x^m + 23400*b ^2*c^2*d^3*m^4*x^6*x^m + 23400*a*b*c*d^4*m^4*x^6*x^m + 2340*a^2*d^5*m^4*x^ 6*x^m + 37520*b^2*c*d^4*m^3*x^7*x^m + 15008*a*b*d^5*m^3*x^7*x^m + 13132*b^ 2*d^5*m^2*x^8*x^m + 2*a*b*c^5*m^7*x^2*x^m + 5*a^2*c^4*d*m^7*x^2*x^m + 33*b ^2*c^5*m^6*x^3*x^m + 330*a*b*c^4*d*m^6*x^3*x^m + 330*a^2*c^3*d^2*m^6*x^3*x ^m + 2090*b^2*c^4*d*m^5*x^4*x^m + 8360*a*b*c^3*d^2*m^5*x^4*x^m + 4180*a...
Time = 1.12 (sec) , antiderivative size = 781, normalized size of antiderivative = 3.38 \[ \int x^m (a+b x)^2 (c+d x)^5 \, dx=\frac {b^2\,d^5\,x^m\,x^8\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {c^3\,x^m\,x^3\,\left (10\,a^2\,d^2+10\,a\,b\,c\,d+b^2\,c^2\right )\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {d^3\,x^m\,x^6\,\left (a^2\,d^2+10\,a\,b\,c\,d+10\,b^2\,c^2\right )\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a^2\,c^5\,x\,x^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a\,c^4\,x^m\,x^2\,\left (5\,a\,d+2\,b\,c\right )\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {b\,d^4\,x^m\,x^7\,\left (2\,a\,d+5\,b\,c\right )\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {5\,c\,d^2\,x^m\,x^5\,\left (a^2\,d^2+4\,a\,b\,c\,d+2\,b^2\,c^2\right )\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {5\,c^2\,d\,x^m\,x^4\,\left (2\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \]
(b^2*d^5*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) + (c^3*x^m*x^3*(10*a^2*d^2 + b^2*c^2 + 10*a*b*c*d)*(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) + (d^3*x^m*x^6*(a^2*d^2 + 10*b^2*c^2 + 10*a*b*c*d)*(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) + (a^2*c^5*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 + 1181 24*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320 ) + (a*c^4*x^m*x^2*(5*a*d + 2*b*c)*(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 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (b*d^4*x^m*x^7 *(2*a*d + 5*b*c)*(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) + (5*c*d^2*x^m*x^5*(a^2*d^2 + 2*b^2* c^2 + 4*a*b*c*d)*(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 + 22449*m^4 + 4536* m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (5*c^2*d*x^m*x^4*(2*a^2*d^2 + b...