3.9.42 \(\int x^m (d+e x) (1+2 x+x^2)^5 \, dx\) [842]

3.9.42.1 Optimal result
3.9.42.2 Mathematica [A] (verified)
3.9.42.3 Rubi [A] (verified)
3.9.42.4 Maple [B] (verified)
3.9.42.5 Fricas [B] (verification not implemented)
3.9.42.6 Sympy [B] (verification not implemented)
3.9.42.7 Maxima [A] (verification not implemented)
3.9.42.8 Giac [B] (verification not implemented)
3.9.42.9 Mupad [B] (verification not implemented)

3.9.42.1 Optimal result

Integrand size = 19, antiderivative size = 209 \[ \int x^m (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^{1+m}}{1+m}+\frac {(10 d+e) x^{2+m}}{2+m}+\frac {5 (9 d+2 e) x^{3+m}}{3+m}+\frac {15 (8 d+3 e) x^{4+m}}{4+m}+\frac {30 (7 d+4 e) x^{5+m}}{5+m}+\frac {42 (6 d+5 e) x^{6+m}}{6+m}+\frac {42 (5 d+6 e) x^{7+m}}{7+m}+\frac {30 (4 d+7 e) x^{8+m}}{8+m}+\frac {15 (3 d+8 e) x^{9+m}}{9+m}+\frac {5 (2 d+9 e) x^{10+m}}{10+m}+\frac {(d+10 e) x^{11+m}}{11+m}+\frac {e x^{12+m}}{12+m} \]

output
d*x^(1+m)/(1+m)+(10*d+e)*x^(2+m)/(2+m)+5*(9*d+2*e)*x^(3+m)/(3+m)+15*(8*d+3 
*e)*x^(4+m)/(4+m)+30*(7*d+4*e)*x^(5+m)/(5+m)+42*(6*d+5*e)*x^(6+m)/(6+m)+42 
*(5*d+6*e)*x^(7+m)/(7+m)+30*(4*d+7*e)*x^(8+m)/(8+m)+15*(3*d+8*e)*x^(9+m)/( 
9+m)+5*(2*d+9*e)*x^(10+m)/(10+m)+(d+10*e)*x^(11+m)/(11+m)+e*x^(12+m)/(12+m 
)
 
3.9.42.2 Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.65 \[ \int x^m (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{1+m} \left (e (1+x)^{11}+(-e (1+m)+d (12+m)) \left (\frac {1}{1+m}+\frac {10 x}{2+m}+\frac {45 x^2}{3+m}+\frac {120 x^3}{4+m}+\frac {210 x^4}{5+m}+\frac {252 x^5}{6+m}+\frac {210 x^6}{7+m}+\frac {120 x^7}{8+m}+\frac {45 x^8}{9+m}+\frac {10 x^9}{10+m}+\frac {x^{10}}{11+m}\right )\right )}{12+m} \]

input
Integrate[x^m*(d + e*x)*(1 + 2*x + x^2)^5,x]
 
output
(x^(1 + m)*(e*(1 + x)^11 + (-(e*(1 + m)) + d*(12 + m))*((1 + m)^(-1) + (10 
*x)/(2 + m) + (45*x^2)/(3 + m) + (120*x^3)/(4 + m) + (210*x^4)/(5 + m) + ( 
252*x^5)/(6 + m) + (210*x^6)/(7 + m) + (120*x^7)/(8 + m) + (45*x^8)/(9 + m 
) + (10*x^9)/(10 + m) + x^10/(11 + m))))/(12 + m)
 
3.9.42.3 Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1184, 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.

\(\displaystyle \int \left (x^2+2 x+1\right )^5 x^m (d+e x) \, dx\)

\(\Big \downarrow \) 1184

\(\displaystyle \int (x+1)^{10} x^m (d+e x)dx\)

\(\Big \downarrow \) 85

\(\displaystyle \int \left ((10 d+e) x^{m+1}+5 (9 d+2 e) x^{m+2}+15 (8 d+3 e) x^{m+3}+30 (7 d+4 e) x^{m+4}+42 (6 d+5 e) x^{m+5}+42 (5 d+6 e) x^{m+6}+30 (4 d+7 e) x^{m+7}+15 (3 d+8 e) x^{m+8}+5 (2 d+9 e) x^{m+9}+(d+10 e) x^{m+10}+d x^m+e x^{m+11}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(10 d+e) x^{m+2}}{m+2}+\frac {5 (9 d+2 e) x^{m+3}}{m+3}+\frac {15 (8 d+3 e) x^{m+4}}{m+4}+\frac {30 (7 d+4 e) x^{m+5}}{m+5}+\frac {42 (6 d+5 e) x^{m+6}}{m+6}+\frac {42 (5 d+6 e) x^{m+7}}{m+7}+\frac {30 (4 d+7 e) x^{m+8}}{m+8}+\frac {15 (3 d+8 e) x^{m+9}}{m+9}+\frac {5 (2 d+9 e) x^{m+10}}{m+10}+\frac {(d+10 e) x^{m+11}}{m+11}+\frac {d x^{m+1}}{m+1}+\frac {e x^{m+12}}{m+12}\)

input
Int[x^m*(d + e*x)*(1 + 2*x + x^2)^5,x]
 
output
(d*x^(1 + m))/(1 + m) + ((10*d + e)*x^(2 + m))/(2 + m) + (5*(9*d + 2*e)*x^ 
(3 + m))/(3 + m) + (15*(8*d + 3*e)*x^(4 + m))/(4 + m) + (30*(7*d + 4*e)*x^ 
(5 + m))/(5 + m) + (42*(6*d + 5*e)*x^(6 + m))/(6 + m) + (42*(5*d + 6*e)*x^ 
(7 + m))/(7 + m) + (30*(4*d + 7*e)*x^(8 + m))/(8 + m) + (15*(3*d + 8*e)*x^ 
(9 + m))/(9 + m) + (5*(2*d + 9*e)*x^(10 + m))/(10 + m) + ((d + 10*e)*x^(11 
 + m))/(11 + m) + (e*x^(12 + m))/(12 + m)
 

3.9.42.3.1 Defintions of rubi rules used

rule 85
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])
 

rule 1184
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]
 

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 
3.9.42.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2244\) vs. \(2(209)=418\).

Time = 0.26 (sec) , antiderivative size = 2245, normalized size of antiderivative = 10.74

method result size
risch \(\text {Expression too large to display}\) \(2245\)
gosper \(\text {Expression too large to display}\) \(2246\)
parallelrisch \(\text {Expression too large to display}\) \(3093\)

input
int(x^m*(e*x+d)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
 
output
x^m*(e*m^11*x^11+d*m^11*x^10+10*e*m^11*x^10+66*e*m^10*x^11+10*d*m^11*x^9+6 
7*d*m^10*x^10+45*e*m^11*x^9+670*e*m^10*x^10+1925*e*m^9*x^11+45*d*m^11*x^8+ 
680*d*m^10*x^9+1980*d*m^9*x^10+120*e*m^11*x^8+3060*e*m^10*x^9+19800*e*m^9* 
x^10+32670*e*m^8*x^11+120*d*m^11*x^7+3105*d*m^10*x^8+20370*d*m^9*x^9+33990 
*d*m^8*x^10+210*e*m^11*x^7+8280*e*m^10*x^8+91665*e*m^9*x^9+339900*e*m^8*x^ 
10+357423*e*m^7*x^11+210*d*m^11*x^6+8400*d*m^10*x^7+94320*d*m^9*x^8+354000 
*d*m^8*x^9+375573*d*m^7*x^10+252*e*m^11*x^6+14700*e*m^10*x^7+251520*e*m^9* 
x^8+1593000*e*m^8*x^9+3755730*e*m^7*x^10+2637558*e*m^6*x^11+252*d*m^11*x^5 
+14910*d*m^10*x^6+258840*d*m^9*x^7+1660770*d*m^8*x^8+3954630*d*m^7*x^9+279 
5331*d*m^6*x^10+210*e*m^11*x^5+17892*e*m^10*x^6+452970*e*m^9*x^7+4428720*e 
*m^8*x^8+17795835*e*m^7*x^9+27953310*e*m^6*x^10+13339535*e*m^5*x^11+210*d* 
m^11*x^4+18144*d*m^10*x^5+466200*d*m^9*x^6+4621680*d*m^8*x^7+18778905*d*m^ 
7*x^8+29720040*d*m^6*x^9+14241590*d*m^5*x^10+120*e*m^11*x^4+15120*e*m^10*x 
^5+559440*e*m^9*x^6+8087940*e*m^8*x^7+50077080*e*m^7*x^8+133740180*e*m^6*x 
^9+142415900*e*m^5*x^10+45995730*e*m^4*x^11+120*d*m^11*x^3+15330*d*m^10*x^ 
4+575820*d*m^9*x^5+8448300*d*m^8*x^6+52962120*d*m^7*x^7+142688385*d*m^6*x^ 
8+152701910*d*m^5*x^9+49412660*d*m^4*x^10+45*e*m^11*x^3+8760*e*m^10*x^4+47 
9850*e*m^9*x^5+10137960*e*m^8*x^6+92683710*e*m^7*x^7+380502360*e*m^6*x^8+6 
87158595*e*m^5*x^9+494126600*e*m^4*x^10+105258076*e*m^3*x^11+45*d*m^11*x^2 
+8880*d*m^10*x^3+493920*d*m^9*x^4+10599120*d*m^8*x^5+98249130*d*m^7*x^6...
 
3.9.42.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1569 vs. \(2 (209) = 418\).

Time = 0.31 (sec) , antiderivative size = 1569, normalized size of antiderivative = 7.51 \[ \int x^m (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\text {Too large to display} \]

input
integrate(x^m*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")
 
output
((e*m^11 + 66*e*m^10 + 1925*e*m^9 + 32670*e*m^8 + 357423*e*m^7 + 2637558*e 
*m^6 + 13339535*e*m^5 + 45995730*e*m^4 + 105258076*e*m^3 + 150917976*e*m^2 
 + 120543840*e*m + 39916800*e)*x^12 + ((d + 10*e)*m^11 + 67*(d + 10*e)*m^1 
0 + 1980*(d + 10*e)*m^9 + 33990*(d + 10*e)*m^8 + 375573*(d + 10*e)*m^7 + 2 
795331*(d + 10*e)*m^6 + 14241590*(d + 10*e)*m^5 + 49412660*(d + 10*e)*m^4 
+ 113667576*(d + 10*e)*m^3 + 163671552*(d + 10*e)*m^2 + 131172480*(d + 10* 
e)*m + 43545600*d + 435456000*e)*x^11 + 5*((2*d + 9*e)*m^11 + 68*(2*d + 9* 
e)*m^10 + 2037*(2*d + 9*e)*m^9 + 35400*(2*d + 9*e)*m^8 + 395463*(2*d + 9*e 
)*m^7 + 2972004*(2*d + 9*e)*m^6 + 15270191*(2*d + 9*e)*m^5 + 53368240*(2*d 
 + 9*e)*m^4 + 123524436*(2*d + 9*e)*m^3 + 178770528*(2*d + 9*e)*m^2 + 1438 
54272*(2*d + 9*e)*m + 95800320*d + 431101440*e)*x^10 + 15*((3*d + 8*e)*m^1 
1 + 69*(3*d + 8*e)*m^10 + 2096*(3*d + 8*e)*m^9 + 36906*(3*d + 8*e)*m^8 + 4 
17309*(3*d + 8*e)*m^7 + 3170853*(3*d + 8*e)*m^6 + 16452554*(3*d + 8*e)*m^5 
 + 57997164*(3*d + 8*e)*m^4 + 135232360*(3*d + 8*e)*m^3 + 196923648*(3*d + 
 8*e)*m^2 + 159246720*(3*d + 8*e)*m + 159667200*d + 425779200*e)*x^9 + 30* 
((4*d + 7*e)*m^11 + 70*(4*d + 7*e)*m^10 + 2157*(4*d + 7*e)*m^9 + 38514*(4* 
d + 7*e)*m^8 + 441351*(4*d + 7*e)*m^7 + 3395826*(4*d + 7*e)*m^6 + 17823623 
*(4*d + 7*e)*m^5 + 63481166*(4*d + 7*e)*m^4 + 149357508*(4*d + 7*e)*m^3 + 
219154824*(4*d + 7*e)*m^2 + 178320960*(4*d + 7*e)*m + 239500800*d + 419126 
400*e)*x^8 + 42*((5*d + 6*e)*m^11 + 71*(5*d + 6*e)*m^10 + 2220*(5*d + 6...
 
3.9.42.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20971 vs. \(2 (168) = 336\).

Time = 1.52 (sec) , antiderivative size = 20971, normalized size of antiderivative = 100.34 \[ \int x^m (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\text {Too large to display} \]

input
integrate(x**m*(e*x+d)*(x**2+2*x+1)**5,x)
 
output
Piecewise((-d/x - 5*d/x**2 - 15*d/x**3 - 30*d/x**4 - 42*d/x**5 - 42*d/x**6 
 - 30*d/x**7 - 15*d/x**8 - 5*d/x**9 - d/x**10 - d/(11*x**11) + e*log(x) - 
10*e/x - 45*e/(2*x**2) - 40*e/x**3 - 105*e/(2*x**4) - 252*e/(5*x**5) - 35* 
e/x**6 - 120*e/(7*x**7) - 45*e/(8*x**8) - 10*e/(9*x**9) - e/(10*x**10), Eq 
(m, -12)), (d*log(x) - 10*d/x - 45*d/(2*x**2) - 40*d/x**3 - 105*d/(2*x**4) 
 - 252*d/(5*x**5) - 35*d/x**6 - 120*d/(7*x**7) - 45*d/(8*x**8) - 10*d/(9*x 
**9) - d/(10*x**10) + e*x + 10*e*log(x) - 45*e/x - 60*e/x**2 - 70*e/x**3 - 
 63*e/x**4 - 42*e/x**5 - 20*e/x**6 - 45*e/(7*x**7) - 5*e/(4*x**8) - e/(9*x 
**9), Eq(m, -11)), (d*x + 10*d*log(x) - 45*d/x - 60*d/x**2 - 70*d/x**3 - 6 
3*d/x**4 - 42*d/x**5 - 20*d/x**6 - 45*d/(7*x**7) - 5*d/(4*x**8) - d/(9*x** 
9) + e*x**2/2 + 10*e*x + 45*e*log(x) - 120*e/x - 105*e/x**2 - 84*e/x**3 - 
105*e/(2*x**4) - 24*e/x**5 - 15*e/(2*x**6) - 10*e/(7*x**7) - e/(8*x**8), E 
q(m, -10)), (d*x**2/2 + 10*d*x + 45*d*log(x) - 120*d/x - 105*d/x**2 - 84*d 
/x**3 - 105*d/(2*x**4) - 24*d/x**5 - 15*d/(2*x**6) - 10*d/(7*x**7) - d/(8* 
x**8) + e*x**3/3 + 5*e*x**2 + 45*e*x + 120*e*log(x) - 210*e/x - 126*e/x**2 
 - 70*e/x**3 - 30*e/x**4 - 9*e/x**5 - 5*e/(3*x**6) - e/(7*x**7), Eq(m, -9) 
), (d*x**3/3 + 5*d*x**2 + 45*d*x + 120*d*log(x) - 210*d/x - 126*d/x**2 - 7 
0*d/x**3 - 30*d/x**4 - 9*d/x**5 - 5*d/(3*x**6) - d/(7*x**7) + e*x**4/4 + 1 
0*e*x**3/3 + 45*e*x**2/2 + 120*e*x + 210*e*log(x) - 252*e/x - 105*e/x**2 - 
 40*e/x**3 - 45*e/(4*x**4) - 2*e/x**5 - e/(6*x**6), Eq(m, -8)), (d*x**4...
 
3.9.42.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.35 \[ \int x^m (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {e x^{m + 12}}{m + 12} + \frac {d x^{m + 11}}{m + 11} + \frac {10 \, e x^{m + 11}}{m + 11} + \frac {10 \, d x^{m + 10}}{m + 10} + \frac {45 \, e x^{m + 10}}{m + 10} + \frac {45 \, d x^{m + 9}}{m + 9} + \frac {120 \, e x^{m + 9}}{m + 9} + \frac {120 \, d x^{m + 8}}{m + 8} + \frac {210 \, e x^{m + 8}}{m + 8} + \frac {210 \, d x^{m + 7}}{m + 7} + \frac {252 \, e x^{m + 7}}{m + 7} + \frac {252 \, d x^{m + 6}}{m + 6} + \frac {210 \, e x^{m + 6}}{m + 6} + \frac {210 \, d x^{m + 5}}{m + 5} + \frac {120 \, e x^{m + 5}}{m + 5} + \frac {120 \, d x^{m + 4}}{m + 4} + \frac {45 \, e x^{m + 4}}{m + 4} + \frac {45 \, d x^{m + 3}}{m + 3} + \frac {10 \, e x^{m + 3}}{m + 3} + \frac {10 \, d x^{m + 2}}{m + 2} + \frac {e x^{m + 2}}{m + 2} + \frac {d x^{m + 1}}{m + 1} \]

input
integrate(x^m*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")
 
output
e*x^(m + 12)/(m + 12) + d*x^(m + 11)/(m + 11) + 10*e*x^(m + 11)/(m + 11) + 
 10*d*x^(m + 10)/(m + 10) + 45*e*x^(m + 10)/(m + 10) + 45*d*x^(m + 9)/(m + 
 9) + 120*e*x^(m + 9)/(m + 9) + 120*d*x^(m + 8)/(m + 8) + 210*e*x^(m + 8)/ 
(m + 8) + 210*d*x^(m + 7)/(m + 7) + 252*e*x^(m + 7)/(m + 7) + 252*d*x^(m + 
 6)/(m + 6) + 210*e*x^(m + 6)/(m + 6) + 210*d*x^(m + 5)/(m + 5) + 120*e*x^ 
(m + 5)/(m + 5) + 120*d*x^(m + 4)/(m + 4) + 45*e*x^(m + 4)/(m + 4) + 45*d* 
x^(m + 3)/(m + 3) + 10*e*x^(m + 3)/(m + 3) + 10*d*x^(m + 2)/(m + 2) + e*x^ 
(m + 2)/(m + 2) + d*x^(m + 1)/(m + 1)
 
3.9.42.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3092 vs. \(2 (209) = 418\).

Time = 0.32 (sec) , antiderivative size = 3092, normalized size of antiderivative = 14.79 \[ \int x^m (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\text {Too large to display} \]

input
integrate(x^m*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")
 
output
(e*m^11*x^12*x^m + d*m^11*x^11*x^m + 10*e*m^11*x^11*x^m + 66*e*m^10*x^12*x 
^m + 10*d*m^11*x^10*x^m + 45*e*m^11*x^10*x^m + 67*d*m^10*x^11*x^m + 670*e* 
m^10*x^11*x^m + 1925*e*m^9*x^12*x^m + 45*d*m^11*x^9*x^m + 120*e*m^11*x^9*x 
^m + 680*d*m^10*x^10*x^m + 3060*e*m^10*x^10*x^m + 1980*d*m^9*x^11*x^m + 19 
800*e*m^9*x^11*x^m + 32670*e*m^8*x^12*x^m + 120*d*m^11*x^8*x^m + 210*e*m^1 
1*x^8*x^m + 3105*d*m^10*x^9*x^m + 8280*e*m^10*x^9*x^m + 20370*d*m^9*x^10*x 
^m + 91665*e*m^9*x^10*x^m + 33990*d*m^8*x^11*x^m + 339900*e*m^8*x^11*x^m + 
 357423*e*m^7*x^12*x^m + 210*d*m^11*x^7*x^m + 252*e*m^11*x^7*x^m + 8400*d* 
m^10*x^8*x^m + 14700*e*m^10*x^8*x^m + 94320*d*m^9*x^9*x^m + 251520*e*m^9*x 
^9*x^m + 354000*d*m^8*x^10*x^m + 1593000*e*m^8*x^10*x^m + 375573*d*m^7*x^1 
1*x^m + 3755730*e*m^7*x^11*x^m + 2637558*e*m^6*x^12*x^m + 252*d*m^11*x^6*x 
^m + 210*e*m^11*x^6*x^m + 14910*d*m^10*x^7*x^m + 17892*e*m^10*x^7*x^m + 25 
8840*d*m^9*x^8*x^m + 452970*e*m^9*x^8*x^m + 1660770*d*m^8*x^9*x^m + 442872 
0*e*m^8*x^9*x^m + 3954630*d*m^7*x^10*x^m + 17795835*e*m^7*x^10*x^m + 27953 
31*d*m^6*x^11*x^m + 27953310*e*m^6*x^11*x^m + 13339535*e*m^5*x^12*x^m + 21 
0*d*m^11*x^5*x^m + 120*e*m^11*x^5*x^m + 18144*d*m^10*x^6*x^m + 15120*e*m^1 
0*x^6*x^m + 466200*d*m^9*x^7*x^m + 559440*e*m^9*x^7*x^m + 4621680*d*m^8*x^ 
8*x^m + 8087940*e*m^8*x^8*x^m + 18778905*d*m^7*x^9*x^m + 50077080*e*m^7*x^ 
9*x^m + 29720040*d*m^6*x^10*x^m + 133740180*e*m^6*x^10*x^m + 14241590*d*m^ 
5*x^11*x^m + 142415900*e*m^5*x^11*x^m + 45995730*e*m^4*x^12*x^m + 120*d...
 
3.9.42.9 Mupad [B] (verification not implemented)

Time = 11.37 (sec) , antiderivative size = 1515, normalized size of antiderivative = 7.25 \[ \int x^m (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\text {Too large to display} \]

input
int(x^m*(d + e*x)*(2*x + x^2 + 1)^5,x)
 
output
(e*x^m*x^12*(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995730*m^4 + 
13339535*m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + 
 m^11 + 39916800))/(1486442880*m + 1931559552*m^2 + 1414014888*m^3 + 65720 
6836*m^4 + 206070150*m^5 + 44990231*m^6 + 6926634*m^7 + 749463*m^8 + 55770 
*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600) + (x^m*x^11*(d + 10*e)*(131 
172480*m + 163671552*m^2 + 113667576*m^3 + 49412660*m^4 + 14241590*m^5 + 2 
795331*m^6 + 375573*m^7 + 33990*m^8 + 1980*m^9 + 67*m^10 + m^11 + 43545600 
))/(1486442880*m + 1931559552*m^2 + 1414014888*m^3 + 657206836*m^4 + 20607 
0150*m^5 + 44990231*m^6 + 6926634*m^7 + 749463*m^8 + 55770*m^9 + 2717*m^10 
 + 78*m^11 + m^12 + 479001600) + (d*x*x^m*(1007441280*m + 924118272*m^2 + 
489896616*m^3 + 167310220*m^4 + 38759930*m^5 + 6230301*m^6 + 696333*m^7 + 
53130*m^8 + 2640*m^9 + 77*m^10 + m^11 + 479001600))/(1486442880*m + 193155 
9552*m^2 + 1414014888*m^3 + 657206836*m^4 + 206070150*m^5 + 44990231*m^6 + 
 6926634*m^7 + 749463*m^8 + 55770*m^9 + 2717*m^10 + 78*m^11 + m^12 + 47900 
1600) + (x^m*x^2*(10*d + e)*(623471040*m + 654044256*m^2 + 379985316*m^3 + 
 138610760*m^4 + 33729695*m^5 + 5630268*m^6 + 648183*m^7 + 50640*m^8 + 256 
5*m^9 + 76*m^10 + m^11 + 239500800))/(1486442880*m + 1931559552*m^2 + 1414 
014888*m^3 + 657206836*m^4 + 206070150*m^5 + 44990231*m^6 + 6926634*m^7 + 
749463*m^8 + 55770*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600) + (5*x^m* 
x^10*(2*d + 9*e)*(143854272*m + 178770528*m^2 + 123524436*m^3 + 5336824...