Integrand size = 19, antiderivative size = 131 \[ \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=-\frac {1}{11} (d-e) (1+x)^{11}+\frac {1}{12} (7 d-8 e) (1+x)^{12}-\frac {7}{13} (3 d-4 e) (1+x)^{13}+\frac {1}{2} (5 d-8 e) (1+x)^{14}-\frac {7}{3} (d-2 e) (1+x)^{15}+\frac {7}{16} (3 d-8 e) (1+x)^{16}-\frac {7}{17} (d-4 e) (1+x)^{17}+\frac {1}{18} (d-8 e) (1+x)^{18}+\frac {1}{19} e (1+x)^{19} \] Output:
-1/11*(d-e)*(1+x)^11+1/12*(7*d-8*e)*(1+x)^12-7/13*(3*d-4*e)*(1+x)^13+1/2*( 5*d-8*e)*(1+x)^14-7/3*(d-2*e)*(1+x)^15+7/16*(3*d-8*e)*(1+x)^16-7/17*(d-4*e )*(1+x)^17+1/18*(d-8*e)*(1+x)^18+1/19*e*(1+x)^19
Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.14 \[ \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^8}{8}+\frac {1}{9} (10 d+e) x^9+\frac {1}{2} (9 d+2 e) x^{10}+\frac {15}{11} (8 d+3 e) x^{11}+\frac {5}{2} (7 d+4 e) x^{12}+\frac {42}{13} (6 d+5 e) x^{13}+3 (5 d+6 e) x^{14}+2 (4 d+7 e) x^{15}+\frac {15}{16} (3 d+8 e) x^{16}+\frac {5}{17} (2 d+9 e) x^{17}+\frac {1}{18} (d+10 e) x^{18}+\frac {e x^{19}}{19} \] Input:
Integrate[x^7*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
(d*x^8)/8 + ((10*d + e)*x^9)/9 + ((9*d + 2*e)*x^10)/2 + (15*(8*d + 3*e)*x^ 11)/11 + (5*(7*d + 4*e)*x^12)/2 + (42*(6*d + 5*e)*x^13)/13 + 3*(5*d + 6*e) *x^14 + 2*(4*d + 7*e)*x^15 + (15*(3*d + 8*e)*x^16)/16 + (5*(2*d + 9*e)*x^1 7)/17 + ((d + 10*e)*x^18)/18 + (e*x^19)/19
Time = 0.57 (sec) , antiderivative size = 131, 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.
| \(\displaystyle \int x^7 \left (x^2+2 x+1\right )^5 (d+e x) \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \int x^7 (x+1)^{10} (d+e x)dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left ((x+1)^{17} (d-8 e)-7 (x+1)^{16} (d-4 e)+7 (x+1)^{15} (3 d-8 e)-35 (x+1)^{14} (d-2 e)+7 (x+1)^{13} (5 d-8 e)-7 (x+1)^{12} (3 d-4 e)+(x+1)^{11} (7 d-8 e)+(x+1)^{10} (e-d)+e (x+1)^{18}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{18} (x+1)^{18} (d-8 e)-\frac {7}{17} (x+1)^{17} (d-4 e)+\frac {7}{16} (x+1)^{16} (3 d-8 e)-\frac {7}{3} (x+1)^{15} (d-2 e)+\frac {1}{2} (x+1)^{14} (5 d-8 e)-\frac {7}{13} (x+1)^{13} (3 d-4 e)+\frac {1}{12} (x+1)^{12} (7 d-8 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{19} e (x+1)^{19}\) |
Input:
Int[x^7*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
-1/11*((d - e)*(1 + x)^11) + ((7*d - 8*e)*(1 + x)^12)/12 - (7*(3*d - 4*e)* (1 + x)^13)/13 + ((5*d - 8*e)*(1 + x)^14)/2 - (7*(d - 2*e)*(1 + x)^15)/3 + (7*(3*d - 8*e)*(1 + x)^16)/16 - (7*(d - 4*e)*(1 + x)^17)/17 + ((d - 8*e)* (1 + x)^18)/18 + (e*(1 + x)^19)/19
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 = 0.82 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.93
| method | result | size |
| norman | \(\frac {d \,x^{8}}{8}+\left (\frac {10 d}{9}+\frac {e}{9}\right ) x^{9}+\left (\frac {9 d}{2}+e \right ) x^{10}+\left (\frac {120 d}{11}+\frac {45 e}{11}\right ) x^{11}+\left (\frac {35 d}{2}+10 e \right ) x^{12}+\left (\frac {252 d}{13}+\frac {210 e}{13}\right ) x^{13}+\left (15 d +18 e \right ) x^{14}+\left (8 d +14 e \right ) x^{15}+\left (\frac {45 d}{16}+\frac {15 e}{2}\right ) x^{16}+\left (\frac {10 d}{17}+\frac {45 e}{17}\right ) x^{17}+\left (\frac {d}{18}+\frac {5 e}{9}\right ) x^{18}+\frac {e \,x^{19}}{19}\) | \(122\) |
| default | \(\frac {e \,x^{19}}{19}+\frac {\left (d +10 e \right ) x^{18}}{18}+\frac {\left (10 d +45 e \right ) x^{17}}{17}+\frac {\left (45 d +120 e \right ) x^{16}}{16}+\frac {\left (120 d +210 e \right ) x^{15}}{15}+\frac {\left (210 d +252 e \right ) x^{14}}{14}+\frac {\left (252 d +210 e \right ) x^{13}}{13}+\frac {\left (210 d +120 e \right ) x^{12}}{12}+\frac {\left (120 d +45 e \right ) x^{11}}{11}+\frac {\left (45 d +10 e \right ) x^{10}}{10}+\frac {\left (10 d +e \right ) x^{9}}{9}+\frac {d \,x^{8}}{8}\) | \(130\) |
| gosper | \(\frac {x^{8} \left (350064 e \,x^{11}+369512 d \,x^{10}+3695120 e \,x^{10}+3912480 d \,x^{9}+17606160 e \,x^{9}+18706545 d \,x^{8}+49884120 e \,x^{8}+53209728 d \,x^{7}+93117024 e \,x^{7}+99768240 d \,x^{6}+119721888 e \,x^{6}+128931264 d \,x^{5}+107442720 x^{5} e +116396280 d \,x^{4}+66512160 x^{4} e +72558720 d \,x^{3}+27209520 x^{3} e +29930472 d \,x^{2}+6651216 e \,x^{2}+7390240 d x +739024 e x +831402 d \right )}{6651216}\) | \(132\) |
| risch | \(\frac {1}{19} e \,x^{19}+\frac {1}{18} x^{18} d +\frac {5}{9} e \,x^{18}+\frac {10}{17} d \,x^{17}+\frac {45}{17} x^{17} e +\frac {45}{16} d \,x^{16}+\frac {15}{2} x^{16} e +8 x^{15} d +14 x^{15} e +15 x^{14} d +18 x^{14} e +\frac {252}{13} d \,x^{13}+\frac {210}{13} x^{13} e +\frac {35}{2} d \,x^{12}+10 x^{12} e +\frac {120}{11} x^{11} d +\frac {45}{11} e \,x^{11}+\frac {9}{2} d \,x^{10}+e \,x^{10}+\frac {10}{9} d \,x^{9}+\frac {1}{9} e \,x^{9}+\frac {1}{8} d \,x^{8}\) | \(133\) |
| parallelrisch | \(\frac {1}{19} e \,x^{19}+\frac {1}{18} x^{18} d +\frac {5}{9} e \,x^{18}+\frac {10}{17} d \,x^{17}+\frac {45}{17} x^{17} e +\frac {45}{16} d \,x^{16}+\frac {15}{2} x^{16} e +8 x^{15} d +14 x^{15} e +15 x^{14} d +18 x^{14} e +\frac {252}{13} d \,x^{13}+\frac {210}{13} x^{13} e +\frac {35}{2} d \,x^{12}+10 x^{12} e +\frac {120}{11} x^{11} d +\frac {45}{11} e \,x^{11}+\frac {9}{2} d \,x^{10}+e \,x^{10}+\frac {10}{9} d \,x^{9}+\frac {1}{9} e \,x^{9}+\frac {1}{8} d \,x^{8}\) | \(133\) |
| orering | \(\frac {x^{8} \left (350064 e \,x^{11}+369512 d \,x^{10}+3695120 e \,x^{10}+3912480 d \,x^{9}+17606160 e \,x^{9}+18706545 d \,x^{8}+49884120 e \,x^{8}+53209728 d \,x^{7}+93117024 e \,x^{7}+99768240 d \,x^{6}+119721888 e \,x^{6}+128931264 d \,x^{5}+107442720 x^{5} e +116396280 d \,x^{4}+66512160 x^{4} e +72558720 d \,x^{3}+27209520 x^{3} e +29930472 d \,x^{2}+6651216 e \,x^{2}+7390240 d x +739024 e x +831402 d \right ) \left (x^{2}+2 x +1\right )^{5}}{6651216 \left (x +1\right )^{10}}\) | \(147\) |
Input:
int(x^7*(e*x+d)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/8*d*x^8+(10/9*d+1/9*e)*x^9+(9/2*d+e)*x^10+(120/11*d+45/11*e)*x^11+(35/2* d+10*e)*x^12+(252/13*d+210/13*e)*x^13+(15*d+18*e)*x^14+(8*d+14*e)*x^15+(45 /16*d+15/2*e)*x^16+(10/17*d+45/17*e)*x^17+(1/18*d+5/9*e)*x^18+1/19*e*x^19
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{19} \, e x^{19} + \frac {1}{18} \, {\left (d + 10 \, e\right )} x^{18} + \frac {5}{17} \, {\left (2 \, d + 9 \, e\right )} x^{17} + \frac {15}{16} \, {\left (3 \, d + 8 \, e\right )} x^{16} + 2 \, {\left (4 \, d + 7 \, e\right )} x^{15} + 3 \, {\left (5 \, d + 6 \, e\right )} x^{14} + \frac {42}{13} \, {\left (6 \, d + 5 \, e\right )} x^{13} + \frac {5}{2} \, {\left (7 \, d + 4 \, e\right )} x^{12} + \frac {15}{11} \, {\left (8 \, d + 3 \, e\right )} x^{11} + \frac {1}{2} \, {\left (9 \, d + 2 \, e\right )} x^{10} + \frac {1}{9} \, {\left (10 \, d + e\right )} x^{9} + \frac {1}{8} \, d x^{8} \] Input:
integrate(x^7*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/19*e*x^19 + 1/18*(d + 10*e)*x^18 + 5/17*(2*d + 9*e)*x^17 + 15/16*(3*d + 8*e)*x^16 + 2*(4*d + 7*e)*x^15 + 3*(5*d + 6*e)*x^14 + 42/13*(6*d + 5*e)*x^ 13 + 5/2*(7*d + 4*e)*x^12 + 15/11*(8*d + 3*e)*x^11 + 1/2*(9*d + 2*e)*x^10 + 1/9*(10*d + e)*x^9 + 1/8*d*x^8
Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^{8}}{8} + \frac {e x^{19}}{19} + x^{18} \left (\frac {d}{18} + \frac {5 e}{9}\right ) + x^{17} \cdot \left (\frac {10 d}{17} + \frac {45 e}{17}\right ) + x^{16} \cdot \left (\frac {45 d}{16} + \frac {15 e}{2}\right ) + x^{15} \cdot \left (8 d + 14 e\right ) + x^{14} \cdot \left (15 d + 18 e\right ) + x^{13} \cdot \left (\frac {252 d}{13} + \frac {210 e}{13}\right ) + x^{12} \cdot \left (\frac {35 d}{2} + 10 e\right ) + x^{11} \cdot \left (\frac {120 d}{11} + \frac {45 e}{11}\right ) + x^{10} \cdot \left (\frac {9 d}{2} + e\right ) + x^{9} \cdot \left (\frac {10 d}{9} + \frac {e}{9}\right ) \] Input:
integrate(x**7*(e*x+d)*(x**2+2*x+1)**5,x)
Output:
d*x**8/8 + e*x**19/19 + x**18*(d/18 + 5*e/9) + x**17*(10*d/17 + 45*e/17) + x**16*(45*d/16 + 15*e/2) + x**15*(8*d + 14*e) + x**14*(15*d + 18*e) + x** 13*(252*d/13 + 210*e/13) + x**12*(35*d/2 + 10*e) + x**11*(120*d/11 + 45*e/ 11) + x**10*(9*d/2 + e) + x**9*(10*d/9 + e/9)
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{19} \, e x^{19} + \frac {1}{18} \, {\left (d + 10 \, e\right )} x^{18} + \frac {5}{17} \, {\left (2 \, d + 9 \, e\right )} x^{17} + \frac {15}{16} \, {\left (3 \, d + 8 \, e\right )} x^{16} + 2 \, {\left (4 \, d + 7 \, e\right )} x^{15} + 3 \, {\left (5 \, d + 6 \, e\right )} x^{14} + \frac {42}{13} \, {\left (6 \, d + 5 \, e\right )} x^{13} + \frac {5}{2} \, {\left (7 \, d + 4 \, e\right )} x^{12} + \frac {15}{11} \, {\left (8 \, d + 3 \, e\right )} x^{11} + \frac {1}{2} \, {\left (9 \, d + 2 \, e\right )} x^{10} + \frac {1}{9} \, {\left (10 \, d + e\right )} x^{9} + \frac {1}{8} \, d x^{8} \] Input:
integrate(x^7*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/19*e*x^19 + 1/18*(d + 10*e)*x^18 + 5/17*(2*d + 9*e)*x^17 + 15/16*(3*d + 8*e)*x^16 + 2*(4*d + 7*e)*x^15 + 3*(5*d + 6*e)*x^14 + 42/13*(6*d + 5*e)*x^ 13 + 5/2*(7*d + 4*e)*x^12 + 15/11*(8*d + 3*e)*x^11 + 1/2*(9*d + 2*e)*x^10 + 1/9*(10*d + e)*x^9 + 1/8*d*x^8
Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.01 \[ \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{19} \, e x^{19} + \frac {1}{18} \, d x^{18} + \frac {5}{9} \, e x^{18} + \frac {10}{17} \, d x^{17} + \frac {45}{17} \, e x^{17} + \frac {45}{16} \, d x^{16} + \frac {15}{2} \, e x^{16} + 8 \, d x^{15} + 14 \, e x^{15} + 15 \, d x^{14} + 18 \, e x^{14} + \frac {252}{13} \, d x^{13} + \frac {210}{13} \, e x^{13} + \frac {35}{2} \, d x^{12} + 10 \, e x^{12} + \frac {120}{11} \, d x^{11} + \frac {45}{11} \, e x^{11} + \frac {9}{2} \, d x^{10} + e x^{10} + \frac {10}{9} \, d x^{9} + \frac {1}{9} \, e x^{9} + \frac {1}{8} \, d x^{8} \] Input:
integrate(x^7*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/19*e*x^19 + 1/18*d*x^18 + 5/9*e*x^18 + 10/17*d*x^17 + 45/17*e*x^17 + 45/ 16*d*x^16 + 15/2*e*x^16 + 8*d*x^15 + 14*e*x^15 + 15*d*x^14 + 18*e*x^14 + 2 52/13*d*x^13 + 210/13*e*x^13 + 35/2*d*x^12 + 10*e*x^12 + 120/11*d*x^11 + 4 5/11*e*x^11 + 9/2*d*x^10 + e*x^10 + 10/9*d*x^9 + 1/9*e*x^9 + 1/8*d*x^8
Time = 10.96 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {e\,x^{19}}{19}+\left (\frac {d}{18}+\frac {5\,e}{9}\right )\,x^{18}+\left (\frac {10\,d}{17}+\frac {45\,e}{17}\right )\,x^{17}+\left (\frac {45\,d}{16}+\frac {15\,e}{2}\right )\,x^{16}+\left (8\,d+14\,e\right )\,x^{15}+\left (15\,d+18\,e\right )\,x^{14}+\left (\frac {252\,d}{13}+\frac {210\,e}{13}\right )\,x^{13}+\left (\frac {35\,d}{2}+10\,e\right )\,x^{12}+\left (\frac {120\,d}{11}+\frac {45\,e}{11}\right )\,x^{11}+\left (\frac {9\,d}{2}+e\right )\,x^{10}+\left (\frac {10\,d}{9}+\frac {e}{9}\right )\,x^9+\frac {d\,x^8}{8} \] Input:
int(x^7*(d + e*x)*(2*x + x^2 + 1)^5,x)
Output:
x^15*(8*d + 14*e) + x^9*((10*d)/9 + e/9) + x^14*(15*d + 18*e) + x^18*(d/18 + (5*e)/9) + x^12*((35*d)/2 + 10*e) + x^16*((45*d)/16 + (15*e)/2) + x^17* ((10*d)/17 + (45*e)/17) + x^11*((120*d)/11 + (45*e)/11) + x^13*((252*d)/13 + (210*e)/13) + (d*x^8)/8 + (e*x^19)/19 + x^10*((9*d)/2 + e)
Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{8} \left (350064 e \,x^{11}+369512 d \,x^{10}+3695120 e \,x^{10}+3912480 d \,x^{9}+17606160 e \,x^{9}+18706545 d \,x^{8}+49884120 e \,x^{8}+53209728 d \,x^{7}+93117024 e \,x^{7}+99768240 d \,x^{6}+119721888 e \,x^{6}+128931264 d \,x^{5}+107442720 e \,x^{5}+116396280 d \,x^{4}+66512160 e \,x^{4}+72558720 d \,x^{3}+27209520 e \,x^{3}+29930472 d \,x^{2}+6651216 e \,x^{2}+7390240 d x +739024 e x +831402 d \right )}{6651216} \] Input:
int(x^7*(e*x+d)*(x^2+2*x+1)^5,x)
Output:
(x**8*(369512*d*x**10 + 3912480*d*x**9 + 18706545*d*x**8 + 53209728*d*x**7 + 99768240*d*x**6 + 128931264*d*x**5 + 116396280*d*x**4 + 72558720*d*x**3 + 29930472*d*x**2 + 7390240*d*x + 831402*d + 350064*e*x**11 + 3695120*e*x **10 + 17606160*e*x**9 + 49884120*e*x**8 + 93117024*e*x**7 + 119721888*e*x **6 + 107442720*e*x**5 + 66512160*e*x**4 + 27209520*e*x**3 + 6651216*e*x** 2 + 739024*e*x))/6651216