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 )
Time = 0.55 (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)
Time = 0.64 (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)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(2244\) vs. \(2(209)=418\).
Time = 0.88 (sec) , antiderivative size = 2245, normalized size of antiderivative = 10.74
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(2245\) |
| gosper | \(\text {Expression too large to display}\) | \(2246\) |
| orering | \(\text {Expression too large to display}\) | \(2260\) |
| 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...
Leaf count of result is larger than twice the leaf count of optimal. 1569 vs. \(2 (209) = 418\).
Time = 0.09 (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...
Leaf count of result is larger than twice the leaf count of optimal. 20971 vs. \(2 (168) = 336\).
Time = 1.59 (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...
Time = 0.03 (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)
Leaf count of result is larger than twice the leaf count of optimal. 3092 vs. \(2 (209) = 418\).
Time = 0.27 (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...
Time = 12.17 (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...
Time = 0.28 (sec) , antiderivative size = 2244, normalized size of antiderivative = 10.74 \[ \int x^m (d+e x) \left (1+2 x+x^2\right )^5 \, dx =\text {Too large to display} \] Input:
int(x^m*(e*x+d)*(x^2+2*x+1)^5,x)
Output:
(x**m*x*(d*m**11*x**10 + 10*d*m**11*x**9 + 45*d*m**11*x**8 + 120*d*m**11*x **7 + 210*d*m**11*x**6 + 252*d*m**11*x**5 + 210*d*m**11*x**4 + 120*d*m**11 *x**3 + 45*d*m**11*x**2 + 10*d*m**11*x + d*m**11 + 67*d*m**10*x**10 + 680* d*m**10*x**9 + 3105*d*m**10*x**8 + 8400*d*m**10*x**7 + 14910*d*m**10*x**6 + 18144*d*m**10*x**5 + 15330*d*m**10*x**4 + 8880*d*m**10*x**3 + 3375*d*m** 10*x**2 + 760*d*m**10*x + 77*d*m**10 + 1980*d*m**9*x**10 + 20370*d*m**9*x* *9 + 94320*d*m**9*x**8 + 258840*d*m**9*x**7 + 466200*d*m**9*x**6 + 575820* d*m**9*x**5 + 493920*d*m**9*x**4 + 290520*d*m**9*x**3 + 112140*d*m**9*x**2 + 25650*d*m**9*x + 2640*d*m**9 + 33990*d*m**8*x**10 + 354000*d*m**8*x**9 + 1660770*d*m**8*x**8 + 4621680*d*m**8*x**7 + 8448300*d*m**8*x**6 + 105991 20*d*m**8*x**5 + 9242100*d*m**8*x**4 + 5530320*d*m**8*x**3 + 2173230*d*m** 8*x**2 + 506400*d*m**8*x + 53130*d*m**8 + 375573*d*m**7*x**10 + 3954630*d* m**7*x**9 + 18778905*d*m**7*x**8 + 52962120*d*m**7*x**7 + 98249130*d*m**7* x**6 + 125269956*d*m**7*x**5 + 111176730*d*m**7*x**4 + 67814280*d*m**7*x** 3 + 27206145*d*m**7*x**2 + 6481830*d*m**7*x + 696333*d*m**7 + 2795331*d*m* *6*x**10 + 29720040*d*m**6*x**9 + 142688385*d*m**6*x**8 + 407499120*d*m**6 *x**7 + 766849230*d*m**6*x**6 + 993892032*d*m**6*x**5 + 898709490*d*m**6*x **4 + 559938960*d*m**6*x**3 + 230080095*d*m**6*x**2 + 56302680*d*m**6*x + 6230301*d*m**6 + 14241590*d*m**5*x**10 + 152701910*d*m**5*x**9 + 740364930 *d*m**5*x**8 + 2138834760*d*m**5*x**7 + 4080003900*d*m**5*x**6 + 537418...