3.842 \(\int x^m (d+e x) (1+2 x+x^2)^5 \, dx\)
Optimal. Leaf size=209 \[ \frac {(10 d+e) x^{m+2}}{m+2}+\frac {5 (9 d+2 e) x^{m+3}}{m+3}+\frac {15 (8 d+3 e) x^{m+4}}{m+4}+\frac {30 (7 d+4 e) x^{m+5}}{m+5}+\frac {42 (6 d+5 e) x^{m+6}}{m+6}+\frac {42 (5 d+6 e) x^{m+7}}{m+7}+\frac {30 (4 d+7 e) x^{m+8}}{m+8}+\frac {15 (3 d+8 e) x^{m+9}}{m+9}+\frac {5 (2 d+9 e) x^{m+10}}{m+10}+\frac {(d+10 e) x^{m+11}}{m+11}+\frac {d x^{m+1}}{m+1}+\frac {e x^{m+12}}{m+12} \]
[Out]
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)
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Rubi [A] time = 0.10, antiderivative size = 209, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used =
{27, 76} \[ \frac {(10 d+e) x^{m+2}}{m+2}+\frac {5 (9 d+2 e) x^{m+3}}{m+3}+\frac {15 (8 d+3 e) x^{m+4}}{m+4}+\frac {30 (7 d+4 e) x^{m+5}}{m+5}+\frac {42 (6 d+5 e) x^{m+6}}{m+6}+\frac {42 (5 d+6 e) x^{m+7}}{m+7}+\frac {30 (4 d+7 e) x^{m+8}}{m+8}+\frac {15 (3 d+8 e) x^{m+9}}{m+9}+\frac {5 (2 d+9 e) x^{m+10}}{m+10}+\frac {(d+10 e) x^{m+11}}{m+11}+\frac {d x^{m+1}}{m+1}+\frac {e x^{m+12}}{m+12} \]
Antiderivative was successfully verified.
[In]
Int[x^m*(d + e*x)*(1 + 2*x + x^2)^5,x]
[Out]
(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)
Rule 27
Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Rule 76
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> 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]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Rubi steps
\begin {align*} \int x^m (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^m (1+x)^{10} (d+e x) \, dx\\ &=\int \left (d x^m+(10 d+e) x^{1+m}+5 (9 d+2 e) x^{2+m}+15 (8 d+3 e) x^{3+m}+30 (7 d+4 e) x^{4+m}+42 (6 d+5 e) x^{5+m}+42 (5 d+6 e) x^{6+m}+30 (4 d+7 e) x^{7+m}+15 (3 d+8 e) x^{8+m}+5 (2 d+9 e) x^{9+m}+(d+10 e) x^{10+m}+e x^{11+m}\right ) \, 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}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 135, normalized size = 0.65 \[ \frac {x^{m+1} \left (\left (\frac {x^{10}}{m+11}+\frac {10 x^9}{m+10}+\frac {45 x^8}{m+9}+\frac {120 x^7}{m+8}+\frac {210 x^6}{m+7}+\frac {252 x^5}{m+6}+\frac {210 x^4}{m+5}+\frac {120 x^3}{m+4}+\frac {45 x^2}{m+3}+\frac {10 x}{m+2}+\frac {1}{m+1}\right ) (d (m+12)-e (m+1))+e (x+1)^{11}\right )}{m+12} \]
Antiderivative was successfully verified.
[In]
Integrate[x^m*(d + e*x)*(1 + 2*x + x^2)^5,x]
[Out]
(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) + (12
0*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)
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fricas [B] time = 0.66, size = 1569, normalized size = 7.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")
[Out]
((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
^10 + 1980*(d + 10*e)*m^9 + 33990*(d + 10*e)*m^8 + 375573*(d + 10*e)*m^7 + 2795331*(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 + 5336824
0*(2*d + 9*e)*m^4 + 123524436*(2*d + 9*e)*m^3 + 178770528*(2*d + 9*e)*m^2 + 143854272*(2*d + 9*e)*m + 95800320
*d + 431101440*e)*x^10 + 15*((3*d + 8*e)*m^11 + 69*(3*d + 8*e)*m^10 + 2096*(3*d + 8*e)*m^9 + 36906*(3*d + 8*e)
*m^8 + 417309*(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 + 419126400*e)*x^8 + 42*((5*d +
6*e)*m^11 + 71*(5*d + 6*e)*m^10 + 2220*(5*d + 6*e)*m^9 + 40230*(5*d + 6*e)*m^8 + 467853*(5*d + 6*e)*m^7 + 3651
663*(5*d + 6*e)*m^6 + 19428590*(5*d + 6*e)*m^5 + 70070020*(5*d + 6*e)*m^4 + 166716696*(5*d + 6*e)*m^3 + 246998
016*(5*d + 6*e)*m^2 + 202573440*(5*d + 6*e)*m + 342144000*d + 410572800*e)*x^7 + 42*((6*d + 5*e)*m^11 + 72*(6*
d + 5*e)*m^10 + 2285*(6*d + 5*e)*m^9 + 42060*(6*d + 5*e)*m^8 + 497103*(6*d + 5*e)*m^7 + 3944016*(6*d + 5*e)*m^
6 + 21326135*(6*d + 5*e)*m^5 + 78113340*(6*d + 5*e)*m^4 + 188526796*(6*d + 5*e)*m^3 + 282854112*(6*d + 5*e)*m^
2 + 234434880*(6*d + 5*e)*m + 479001600*d + 399168000*e)*x^6 + 30*((7*d + 4*e)*m^11 + 73*(7*d + 4*e)*m^10 + 23
52*(7*d + 4*e)*m^9 + 44010*(7*d + 4*e)*m^8 + 529413*(7*d + 4*e)*m^7 + 4279569*(7*d + 4*e)*m^6 + 23592386*(7*d
+ 4*e)*m^5 + 88108220*(7*d + 4*e)*m^4 + 216665736*(7*d + 4*e)*m^3 + 330686208*(7*d + 4*e)*m^2 + 278128512*(7*d
+ 4*e)*m + 670602240*d + 383201280*e)*x^5 + 15*((8*d + 3*e)*m^11 + 74*(8*d + 3*e)*m^10 + 2421*(8*d + 3*e)*m^9
+ 46086*(8*d + 3*e)*m^8 + 565119*(8*d + 3*e)*m^7 + 4666158*(8*d + 3*e)*m^6 + 26325599*(8*d + 3*e)*m^5 + 10076
7754*(8*d + 3*e)*m^4 + 254135820*(8*d + 3*e)*m^3 + 397471608*(8*d + 3*e)*m^2 + 341673120*(8*d + 3*e)*m + 95800
3200*d + 359251200*e)*x^4 + 5*((9*d + 2*e)*m^11 + 75*(9*d + 2*e)*m^10 + 2492*(9*d + 2*e)*m^9 + 48294*(9*d + 2*
e)*m^8 + 604581*(9*d + 2*e)*m^7 + 5112891*(9*d + 2*e)*m^6 + 29651558*(9*d + 2*e)*m^5 + 117115476*(9*d + 2*e)*m
^4 + 305860408*(9*d + 2*e)*m^3 + 496433664*(9*d + 2*e)*m^2 + 442258560*(9*d + 2*e)*m + 1437004800*d + 31933440
0*e)*x^3 + ((10*d + e)*m^11 + 76*(10*d + e)*m^10 + 2565*(10*d + e)*m^9 + 50640*(10*d + e)*m^8 + 648183*(10*d +
e)*m^7 + 5630268*(10*d + e)*m^6 + 33729695*(10*d + e)*m^5 + 138610760*(10*d + e)*m^4 + 379985316*(10*d + e)*m
^3 + 654044256*(10*d + e)*m^2 + 623471040*(10*d + e)*m + 2395008000*d + 239500800*e)*x^2 + (d*m^11 + 77*d*m^10
+ 2640*d*m^9 + 53130*d*m^8 + 696333*d*m^7 + 6230301*d*m^6 + 38759930*d*m^5 + 167310220*d*m^4 + 489896616*d*m^
3 + 924118272*d*m^2 + 1007441280*d*m + 479001600*d)*x)*x^m/(m^12 + 78*m^11 + 2717*m^10 + 55770*m^9 + 749463*m^
8 + 6926634*m^7 + 44990231*m^6 + 206070150*m^5 + 657206836*m^4 + 1414014888*m^3 + 1931559552*m^2 + 1486442880*
m + 479001600)
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giac [B] time = 0.33, size = 3224, normalized size = 15.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")
[Out]
(m^11*x^12*x^m*e + d*m^11*x^11*x^m + 10*m^11*x^11*x^m*e + 66*m^10*x^12*x^m*e + 10*d*m^11*x^10*x^m + 67*d*m^10*
x^11*x^m + 45*m^11*x^10*x^m*e + 670*m^10*x^11*x^m*e + 1925*m^9*x^12*x^m*e + 45*d*m^11*x^9*x^m + 680*d*m^10*x^1
0*x^m + 1980*d*m^9*x^11*x^m + 120*m^11*x^9*x^m*e + 3060*m^10*x^10*x^m*e + 19800*m^9*x^11*x^m*e + 32670*m^8*x^1
2*x^m*e + 120*d*m^11*x^8*x^m + 3105*d*m^10*x^9*x^m + 20370*d*m^9*x^10*x^m + 33990*d*m^8*x^11*x^m + 210*m^11*x^
8*x^m*e + 8280*m^10*x^9*x^m*e + 91665*m^9*x^10*x^m*e + 339900*m^8*x^11*x^m*e + 357423*m^7*x^12*x^m*e + 210*d*m
^11*x^7*x^m + 8400*d*m^10*x^8*x^m + 94320*d*m^9*x^9*x^m + 354000*d*m^8*x^10*x^m + 375573*d*m^7*x^11*x^m + 252*
m^11*x^7*x^m*e + 14700*m^10*x^8*x^m*e + 251520*m^9*x^9*x^m*e + 1593000*m^8*x^10*x^m*e + 3755730*m^7*x^11*x^m*e
+ 2637558*m^6*x^12*x^m*e + 252*d*m^11*x^6*x^m + 14910*d*m^10*x^7*x^m + 258840*d*m^9*x^8*x^m + 1660770*d*m^8*x
^9*x^m + 3954630*d*m^7*x^10*x^m + 2795331*d*m^6*x^11*x^m + 210*m^11*x^6*x^m*e + 17892*m^10*x^7*x^m*e + 452970*
m^9*x^8*x^m*e + 4428720*m^8*x^9*x^m*e + 17795835*m^7*x^10*x^m*e + 27953310*m^6*x^11*x^m*e + 13339535*m^5*x^12*
x^m*e + 210*d*m^11*x^5*x^m + 18144*d*m^10*x^6*x^m + 466200*d*m^9*x^7*x^m + 4621680*d*m^8*x^8*x^m + 18778905*d*
m^7*x^9*x^m + 29720040*d*m^6*x^10*x^m + 14241590*d*m^5*x^11*x^m + 120*m^11*x^5*x^m*e + 15120*m^10*x^6*x^m*e +
559440*m^9*x^7*x^m*e + 8087940*m^8*x^8*x^m*e + 50077080*m^7*x^9*x^m*e + 133740180*m^6*x^10*x^m*e + 142415900*m
^5*x^11*x^m*e + 45995730*m^4*x^12*x^m*e + 120*d*m^11*x^4*x^m + 15330*d*m^10*x^5*x^m + 575820*d*m^9*x^6*x^m + 8
448300*d*m^8*x^7*x^m + 52962120*d*m^7*x^8*x^m + 142688385*d*m^6*x^9*x^m + 152701910*d*m^5*x^10*x^m + 49412660*
d*m^4*x^11*x^m + 45*m^11*x^4*x^m*e + 8760*m^10*x^5*x^m*e + 479850*m^9*x^6*x^m*e + 10137960*m^8*x^7*x^m*e + 926
83710*m^7*x^8*x^m*e + 380502360*m^6*x^9*x^m*e + 687158595*m^5*x^10*x^m*e + 494126600*m^4*x^11*x^m*e + 10525807
6*m^3*x^12*x^m*e + 45*d*m^11*x^3*x^m + 8880*d*m^10*x^4*x^m + 493920*d*m^9*x^5*x^m + 10599120*d*m^8*x^6*x^m + 9
8249130*d*m^7*x^7*x^m + 407499120*d*m^6*x^8*x^m + 740364930*d*m^5*x^9*x^m + 533682400*d*m^4*x^10*x^m + 1136675
76*d*m^3*x^11*x^m + 10*m^11*x^3*x^m*e + 3330*m^10*x^4*x^m*e + 282240*m^9*x^5*x^m*e + 8832600*m^8*x^6*x^m*e + 1
17898956*m^7*x^7*x^m*e + 713123460*m^6*x^8*x^m*e + 1974306480*m^5*x^9*x^m*e + 2401570800*m^4*x^10*x^m*e + 1136
675760*m^3*x^11*x^m*e + 150917976*m^2*x^12*x^m*e + 10*d*m^11*x^2*x^m + 3375*d*m^10*x^3*x^m + 290520*d*m^9*x^4*
x^m + 9242100*d*m^8*x^5*x^m + 125269956*d*m^7*x^6*x^m + 766849230*d*m^6*x^7*x^m + 2138834760*d*m^5*x^8*x^m + 2
609872380*d*m^4*x^9*x^m + 1235244360*d*m^3*x^10*x^m + 163671552*d*m^2*x^11*x^m + m^11*x^2*x^m*e + 750*m^10*x^3
*x^m*e + 108945*m^9*x^4*x^m*e + 5281200*m^8*x^5*x^m*e + 104391630*m^7*x^6*x^m*e + 920219076*m^6*x^7*x^m*e + 37
42960830*m^5*x^8*x^m*e + 6959659680*m^4*x^9*x^m*e + 5558599620*m^3*x^10*x^m*e + 1636715520*m^2*x^11*x^m*e + 12
0543840*m*x^12*x^m*e + d*m^11*x*x^m + 760*d*m^10*x^2*x^m + 112140*d*m^9*x^3*x^m + 5530320*d*m^8*x^4*x^m + 1111
76730*d*m^7*x^5*x^m + 993892032*d*m^6*x^6*x^m + 4080003900*d*m^5*x^7*x^m + 7617739920*d*m^4*x^8*x^m + 60854562
00*d*m^3*x^9*x^m + 1787705280*d*m^2*x^10*x^m + 131172480*d*m*x^11*x^m + 76*m^10*x^2*x^m*e + 24920*m^9*x^3*x^m*
e + 2073870*m^8*x^4*x^m*e + 63529560*m^7*x^5*x^m*e + 828243360*m^6*x^6*x^m*e + 4896004680*m^5*x^7*x^m*e + 1333
1044860*m^4*x^8*x^m*e + 16227883200*m^3*x^9*x^m*e + 8044673760*m^2*x^10*x^m*e + 1311724800*m*x^11*x^m*e + 3991
6800*x^12*x^m*e + 77*d*m^10*x*x^m + 25650*d*m^9*x^2*x^m + 2173230*d*m^8*x^3*x^m + 67814280*d*m^7*x^4*x^m + 898
709490*d*m^6*x^5*x^m + 5374186020*d*m^5*x^6*x^m + 14714704200*d*m^4*x^7*x^m + 17922900960*d*m^3*x^8*x^m + 8861
564160*d*m^2*x^9*x^m + 1438542720*d*m*x^10*x^m + 43545600*d*x^11*x^m + 2565*m^9*x^2*x^m*e + 482940*m^8*x^3*x^m
*e + 25430355*m^7*x^4*x^m*e + 513548280*m^6*x^5*x^m*e + 4478488350*m^5*x^6*x^m*e + 17657645040*m^4*x^7*x^m*e +
31365076680*m^3*x^8*x^m*e + 23630837760*m^2*x^9*x^m*e + 6473442240*m*x^10*x^m*e + 435456000*x^11*x^m*e + 2640
*d*m^9*x*x^m + 506400*d*m^8*x^2*x^m + 27206145*d*m^7*x^3*x^m + 559938960*d*m^6*x^4*x^m + 4954401060*d*m^5*x^5*
x^m + 19684561680*d*m^4*x^6*x^m + 35010506160*d*m^3*x^7*x^m + 26298578880*d*m^2*x^8*x^m + 7166102400*d*m*x^9*x
^m + 479001600*d*x^10*x^m + 50640*m^8*x^2*x^m*e + 6045810*m^7*x^3*x^m*e + 209977110*m^6*x^4*x^m*e + 2831086320
*m^5*x^5*x^m*e + 16403801400*m^4*x^6*x^m*e + 42012607392*m^3*x^7*x^m*e + 46022513040*m^2*x^8*x^m*e + 191096064
00*m*x^9*x^m*e + 2155507200*x^10*x^m*e + 53130*d*m^8*x*x^m + 6481830*d*m^7*x^2*x^m + 230080095*d*m^6*x^3*x^m +
3159071880*d*m^5*x^4*x^m + 18502726200*d*m^4*x^5*x^m + 47508752592*d*m^3*x^6*x^m + 51869583360*d*m^2*x^7*x^m
+ 21398515200*d*m*x^8*x^m + 2395008000*d*x^9*x^m + 648183*m^7*x^2*x^m*e + 51128910*m^6*x^3*x^m*e + 1184651955*
m^5*x^4*x^m*e + 10572986400*m^4*x^5*x^m*e + 39590627160*m^3*x^6*x^m*e + 62243500032*m^2*x^7*x^m*e + 3744740160
0*m*x^8*x^m*e + 6386688000*x^9*x^m*e + 696333*d*m^7*x*x^m + 56302680*d*m^6*x^2*x^m + 1334320110*d*m^5*x^3*x^m
+ 12092130480*d*m^4*x^4*x^m + 45499804560*d*m^3*x^5*x^m + 71279236224*d*m^2*x^6*x^m + 42540422400*d*m*x^7*x^m
+ 7185024000*d*x^8*x^m + 5630268*m^6*x^2*x^m*e + 296515580*m^5*x^3*x^m*e + 4534548930*m^4*x^4*x^m*e + 25999888
320*m^3*x^5*x^m*e + 59399363520*m^2*x^6*x^m*e + 51048506880*m*x^7*x^m*e + 12573792000*x^8*x^m*e + 6230301*d*m^
6*x*x^m + 337296950*d*m^5*x^2*x^m + 5270196420*d*m^4*x^3*x^m + 30496298400*d*m^3*x^4*x^m + 69444103680*d*m^2*x
^5*x^m + 59077589760*d*m*x^6*x^m + 14370048000*d*x^7*x^m + 33729695*m^5*x^2*x^m*e + 1171154760*m^4*x^3*x^m*e +
11436111900*m^3*x^4*x^m*e + 39682344960*m^2*x^5*x^m*e + 49231324800*m*x^6*x^m*e + 17244057600*x^7*x^m*e + 387
59930*d*m^5*x*x^m + 1386107600*d*m^4*x^2*x^m + 13763718360*d*m^3*x^3*x^m + 47696592960*d*m^2*x^4*x^m + 5840698
7520*d*m*x^5*x^m + 20118067200*d*x^6*x^m + 138610760*m^4*x^2*x^m*e + 3058604080*m^3*x^3*x^m*e + 17886222360*m^
2*x^4*x^m*e + 33375421440*m*x^5*x^m*e + 16765056000*x^6*x^m*e + 167310220*d*m^4*x*x^m + 3799853160*d*m^3*x^2*x
^m + 22339514880*d*m^2*x^3*x^m + 41000774400*d*m*x^4*x^m + 20118067200*d*x^5*x^m + 379985316*m^3*x^2*x^m*e + 4
964336640*m^2*x^3*x^m*e + 15375290400*m*x^4*x^m*e + 11496038400*x^5*x^m*e + 489896616*d*m^3*x*x^m + 6540442560
*d*m^2*x^2*x^m + 19901635200*d*m*x^3*x^m + 14370048000*d*x^4*x^m + 654044256*m^2*x^2*x^m*e + 4422585600*m*x^3*
x^m*e + 5388768000*x^4*x^m*e + 924118272*d*m^2*x*x^m + 6234710400*d*m*x^2*x^m + 7185024000*d*x^3*x^m + 6234710
40*m*x^2*x^m*e + 1596672000*x^3*x^m*e + 1007441280*d*m*x*x^m + 2395008000*d*x^2*x^m + 239500800*x^2*x^m*e + 47
9001600*d*x*x^m)/(m^12 + 78*m^11 + 2717*m^10 + 55770*m^9 + 749463*m^8 + 6926634*m^7 + 44990231*m^6 + 206070150
*m^5 + 657206836*m^4 + 1414014888*m^3 + 1931559552*m^2 + 1486442880*m + 479001600)
________________________________________________________________________________________
maple [B] time = 0.05, size = 2246, normalized size = 10.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(e*x+d)*(x^2+2*x+1)^5,x)
[Out]
x^(m+1)*(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+67*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+828
0*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+375573
0*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+2795331*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+1
8778905*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+8087
940*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+1
5330*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+479850*e*m^9*x^5+10137960*e*m^8*x^6+92683710*e*m^7*x^7+38050
2360*e*m^6*x^8+687158595*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+493
920*d*m^9*x^4+10599120*d*m^8*x^5+98249130*d*m^7*x^6+407499120*d*m^6*x^7+740364930*d*m^5*x^8+533682400*d*m^4*x^
9+113667576*d*m^3*x^10+10*e*m^11*x^2+3330*e*m^10*x^3+282240*e*m^9*x^4+8832600*e*m^8*x^5+117898956*e*m^7*x^6+71
3123460*e*m^6*x^7+1974306480*e*m^5*x^8+2401570800*e*m^4*x^9+1136675760*e*m^3*x^10+150917976*e*m^2*x^11+10*d*m^
11*x+3375*d*m^10*x^2+290520*d*m^9*x^3+9242100*d*m^8*x^4+125269956*d*m^7*x^5+766849230*d*m^6*x^6+2138834760*d*m
^5*x^7+2609872380*d*m^4*x^8+1235244360*d*m^3*x^9+163671552*d*m^2*x^10+e*m^11*x+750*e*m^10*x^2+108945*e*m^9*x^3
+5281200*e*m^8*x^4+104391630*e*m^7*x^5+920219076*e*m^6*x^6+3742960830*e*m^5*x^7+6959659680*e*m^4*x^8+555859962
0*e*m^3*x^9+1636715520*e*m^2*x^10+120543840*e*m*x^11+d*m^11+760*d*m^10*x+112140*d*m^9*x^2+5530320*d*m^8*x^3+11
1176730*d*m^7*x^4+993892032*d*m^6*x^5+4080003900*d*m^5*x^6+7617739920*d*m^4*x^7+6085456200*d*m^3*x^8+178770528
0*d*m^2*x^9+131172480*d*m*x^10+76*e*m^10*x+24920*e*m^9*x^2+2073870*e*m^8*x^3+63529560*e*m^7*x^4+828243360*e*m^
6*x^5+4896004680*e*m^5*x^6+13331044860*e*m^4*x^7+16227883200*e*m^3*x^8+8044673760*e*m^2*x^9+1311724800*e*m*x^1
0+39916800*e*x^11+77*d*m^10+25650*d*m^9*x+2173230*d*m^8*x^2+67814280*d*m^7*x^3+898709490*d*m^6*x^4+5374186020*
d*m^5*x^5+14714704200*d*m^4*x^6+17922900960*d*m^3*x^7+8861564160*d*m^2*x^8+1438542720*d*m*x^9+43545600*d*x^10+
2565*e*m^9*x+482940*e*m^8*x^2+25430355*e*m^7*x^3+513548280*e*m^6*x^4+4478488350*e*m^5*x^5+17657645040*e*m^4*x^
6+31365076680*e*m^3*x^7+23630837760*e*m^2*x^8+6473442240*e*m*x^9+435456000*e*x^10+2640*d*m^9+506400*d*m^8*x+27
206145*d*m^7*x^2+559938960*d*m^6*x^3+4954401060*d*m^5*x^4+19684561680*d*m^4*x^5+35010506160*d*m^3*x^6+26298578
880*d*m^2*x^7+7166102400*d*m*x^8+479001600*d*x^9+50640*e*m^8*x+6045810*e*m^7*x^2+209977110*e*m^6*x^3+283108632
0*e*m^5*x^4+16403801400*e*m^4*x^5+42012607392*e*m^3*x^6+46022513040*e*m^2*x^7+19109606400*e*m*x^8+2155507200*e
*x^9+53130*d*m^8+6481830*d*m^7*x+230080095*d*m^6*x^2+3159071880*d*m^5*x^3+18502726200*d*m^4*x^4+47508752592*d*
m^3*x^5+51869583360*d*m^2*x^6+21398515200*d*m*x^7+2395008000*d*x^8+648183*e*m^7*x+51128910*e*m^6*x^2+118465195
5*e*m^5*x^3+10572986400*e*m^4*x^4+39590627160*e*m^3*x^5+62243500032*e*m^2*x^6+37447401600*e*m*x^7+6386688000*e
*x^8+696333*d*m^7+56302680*d*m^6*x+1334320110*d*m^5*x^2+12092130480*d*m^4*x^3+45499804560*d*m^3*x^4+7127923622
4*d*m^2*x^5+42540422400*d*m*x^6+7185024000*d*x^7+5630268*e*m^6*x+296515580*e*m^5*x^2+4534548930*e*m^4*x^3+2599
9888320*e*m^3*x^4+59399363520*e*m^2*x^5+51048506880*e*m*x^6+12573792000*e*x^7+6230301*d*m^6+337296950*d*m^5*x+
5270196420*d*m^4*x^2+30496298400*d*m^3*x^3+69444103680*d*m^2*x^4+59077589760*d*m*x^5+14370048000*d*x^6+3372969
5*e*m^5*x+1171154760*e*m^4*x^2+11436111900*e*m^3*x^3+39682344960*e*m^2*x^4+49231324800*e*m*x^5+17244057600*e*x
^6+38759930*d*m^5+1386107600*d*m^4*x+13763718360*d*m^3*x^2+47696592960*d*m^2*x^3+58406987520*d*m*x^4+201180672
00*d*x^5+138610760*e*m^4*x+3058604080*e*m^3*x^2+17886222360*e*m^2*x^3+33375421440*e*m*x^4+16765056000*e*x^5+16
7310220*d*m^4+3799853160*d*m^3*x+22339514880*d*m^2*x^2+41000774400*d*m*x^3+20118067200*d*x^4+379985316*e*m^3*x
+4964336640*e*m^2*x^2+15375290400*e*m*x^3+11496038400*e*x^4+489896616*d*m^3+6540442560*d*m^2*x+19901635200*d*m
*x^2+14370048000*d*x^3+654044256*e*m^2*x+4422585600*e*m*x^2+5388768000*e*x^3+924118272*d*m^2+6234710400*d*m*x+
7185024000*d*x^2+623471040*e*m*x+1596672000*e*x^2+1007441280*d*m+2395008000*d*x+239500800*e*x+479001600*d)/(m+
12)/(m+11)/(m+10)/(m+9)/(m+8)/(m+7)/(m+6)/(m+5)/(m+4)/(m+3)/(m+2)/(m+1)
________________________________________________________________________________________
maxima [A] time = 0.58, size = 283, normalized size = 1.35 \[ \frac {e x^{m + 12}}{m + 12} + \frac {d x^{m + 11}}{m + 11} + \frac {10 \, e x^{m + 11}}{m + 11} + \frac {10 \, d x^{m + 10}}{m + 10} + \frac {45 \, e x^{m + 10}}{m + 10} + \frac {45 \, d x^{m + 9}}{m + 9} + \frac {120 \, e x^{m + 9}}{m + 9} + \frac {120 \, d x^{m + 8}}{m + 8} + \frac {210 \, e x^{m + 8}}{m + 8} + \frac {210 \, d x^{m + 7}}{m + 7} + \frac {252 \, e x^{m + 7}}{m + 7} + \frac {252 \, d x^{m + 6}}{m + 6} + \frac {210 \, e x^{m + 6}}{m + 6} + \frac {210 \, d x^{m + 5}}{m + 5} + \frac {120 \, e x^{m + 5}}{m + 5} + \frac {120 \, d x^{m + 4}}{m + 4} + \frac {45 \, e x^{m + 4}}{m + 4} + \frac {45 \, d x^{m + 3}}{m + 3} + \frac {10 \, e x^{m + 3}}{m + 3} + \frac {10 \, d x^{m + 2}}{m + 2} + \frac {e x^{m + 2}}{m + 2} + \frac {d x^{m + 1}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")
[Out]
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) + 4
5*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)
________________________________________________________________________________________
mupad [B] time = 2.45, size = 1515, normalized size = 7.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(d + e*x)*(2*x + x^2 + 1)^5,x)
[Out]
(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 + 657
206836*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)*(131172480*m + 163671552*m^2 + 113667576*m^3 + 49412660*m^4 + 14241590*m^5
+ 2795331*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 + 206070150*m^5 + 44990231*m^6 + 6926634*m^7 + 749463*m^8 + 55770*m^9 + 271
7*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
+ 1931559552*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 + 479001600) + (x^m*x^2*(10*d + e)*(623471040*m + 654044256*m^2 + 37998
5316*m^3 + 138610760*m^4 + 33729695*m^5 + 5630268*m^6 + 648183*m^7 + 50640*m^8 + 2565*m^9 + 76*m^10 + m^11 + 2
39500800))/(1486442880*m + 1931559552*m^2 + 1414014888*m^3 + 657206836*m^4 + 206070150*m^5 + 44990231*m^6 + 69
26634*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)*(143854
272*m + 178770528*m^2 + 123524436*m^3 + 53368240*m^4 + 15270191*m^5 + 2972004*m^6 + 395463*m^7 + 35400*m^8 + 2
037*m^9 + 68*m^10 + m^11 + 47900160))/(1486442880*m + 1931559552*m^2 + 1414014888*m^3 + 657206836*m^4 + 206070
150*m^5 + 44990231*m^6 + 6926634*m^7 + 749463*m^8 + 55770*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600) + (15*
x^m*x^9*(3*d + 8*e)*(159246720*m + 196923648*m^2 + 135232360*m^3 + 57997164*m^4 + 16452554*m^5 + 3170853*m^6 +
417309*m^7 + 36906*m^8 + 2096*m^9 + 69*m^10 + m^11 + 53222400))/(1486442880*m + 1931559552*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 + 479001600) + (30*x^m*x^8*(4*d + 7*e)*(178320960*m + 219154824*m^2 + 149357508*m^3 + 63481166*m^4 + 1
7823623*m^5 + 3395826*m^6 + 441351*m^7 + 38514*m^8 + 2157*m^9 + 70*m^10 + m^11 + 59875200))/(1486442880*m + 19
31559552*m^2 + 1414014888*m^3 + 657206836*m^4 + 206070150*m^5 + 44990231*m^6 + 6926634*m^7 + 749463*m^8 + 5577
0*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600) + (42*x^m*x^7*(5*d + 6*e)*(202573440*m + 246998016*m^2 + 16671
6696*m^3 + 70070020*m^4 + 19428590*m^5 + 3651663*m^6 + 467853*m^7 + 40230*m^8 + 2220*m^9 + 71*m^10 + m^11 + 68
428800))/(1486442880*m + 1931559552*m^2 + 1414014888*m^3 + 657206836*m^4 + 206070150*m^5 + 44990231*m^6 + 6926
634*m^7 + 749463*m^8 + 55770*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600) + (42*x^m*x^6*(6*d + 5*e)*(23443488
0*m + 282854112*m^2 + 188526796*m^3 + 78113340*m^4 + 21326135*m^5 + 3944016*m^6 + 497103*m^7 + 42060*m^8 + 228
5*m^9 + 72*m^10 + m^11 + 79833600))/(1486442880*m + 1931559552*m^2 + 1414014888*m^3 + 657206836*m^4 + 20607015
0*m^5 + 44990231*m^6 + 6926634*m^7 + 749463*m^8 + 55770*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600) + (30*x^
m*x^5*(7*d + 4*e)*(278128512*m + 330686208*m^2 + 216665736*m^3 + 88108220*m^4 + 23592386*m^5 + 4279569*m^6 + 5
29413*m^7 + 44010*m^8 + 2352*m^9 + 73*m^10 + m^11 + 95800320))/(1486442880*m + 1931559552*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 + 479001600) + (15*x^m*x^4*(8*d + 3*e)*(341673120*m + 397471608*m^2 + 254135820*m^3 + 100767754*m^4 + 26
325599*m^5 + 4666158*m^6 + 565119*m^7 + 46086*m^8 + 2421*m^9 + 74*m^10 + m^11 + 119750400))/(1486442880*m + 19
31559552*m^2 + 1414014888*m^3 + 657206836*m^4 + 206070150*m^5 + 44990231*m^6 + 6926634*m^7 + 749463*m^8 + 5577
0*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600) + (5*x^m*x^3*(9*d + 2*e)*(442258560*m + 496433664*m^2 + 305860
408*m^3 + 117115476*m^4 + 29651558*m^5 + 5112891*m^6 + 604581*m^7 + 48294*m^8 + 2492*m^9 + 75*m^10 + m^11 + 15
9667200))/(1486442880*m + 1931559552*m^2 + 1414014888*m^3 + 657206836*m^4 + 206070150*m^5 + 44990231*m^6 + 692
6634*m^7 + 749463*m^8 + 55770*m^9 + 2717*m^10 + 78*m^11 + m^12 + 479001600)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m*(e*x+d)*(x**2+2*x+1)**5,x)
[Out]
Timed out
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