Integrand size = 19, antiderivative size = 99 \[ \int x^5 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=-\frac {1}{11} (d-e) (1+x)^{11}+\frac {1}{12} (5 d-6 e) (1+x)^{12}-\frac {5}{13} (2 d-3 e) (1+x)^{13}+\frac {5}{7} (d-2 e) (1+x)^{14}-\frac {1}{3} (d-3 e) (1+x)^{15}+\frac {1}{16} (d-6 e) (1+x)^{16}+\frac {1}{17} e (1+x)^{17} \] Output:
-1/11*(d-e)*(1+x)^11+1/12*(5*d-6*e)*(1+x)^12-5/13*(2*d-3*e)*(1+x)^13+5/7*( d-2*e)*(1+x)^14-1/3*(d-3*e)*(1+x)^15+1/16*(d-6*e)*(1+x)^16+1/17*e*(1+x)^17
Time = 0.02 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.53 \[ \int x^5 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^6}{6}+\frac {1}{7} (10 d+e) x^7+\frac {5}{8} (9 d+2 e) x^8+\frac {5}{3} (8 d+3 e) x^9+3 (7 d+4 e) x^{10}+\frac {42}{11} (6 d+5 e) x^{11}+\frac {7}{2} (5 d+6 e) x^{12}+\frac {30}{13} (4 d+7 e) x^{13}+\frac {15}{14} (3 d+8 e) x^{14}+\frac {1}{3} (2 d+9 e) x^{15}+\frac {1}{16} (d+10 e) x^{16}+\frac {e x^{17}}{17} \] Input:
Integrate[x^5*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
(d*x^6)/6 + ((10*d + e)*x^7)/7 + (5*(9*d + 2*e)*x^8)/8 + (5*(8*d + 3*e)*x^ 9)/3 + 3*(7*d + 4*e)*x^10 + (42*(6*d + 5*e)*x^11)/11 + (7*(5*d + 6*e)*x^12 )/2 + (30*(4*d + 7*e)*x^13)/13 + (15*(3*d + 8*e)*x^14)/14 + ((2*d + 9*e)*x ^15)/3 + ((d + 10*e)*x^16)/16 + (e*x^17)/17
Time = 0.48 (sec) , antiderivative size = 99, 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.
| \(\displaystyle \int x^5 \left (x^2+2 x+1\right )^5 (d+e x) \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \int x^5 (x+1)^{10} (d+e x)dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left ((x+1)^{15} (d-6 e)-5 (x+1)^{14} (d-3 e)+10 (x+1)^{13} (d-2 e)-5 (x+1)^{12} (2 d-3 e)+(x+1)^{11} (5 d-6 e)+(x+1)^{10} (e-d)+e (x+1)^{16}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{16} (x+1)^{16} (d-6 e)-\frac {1}{3} (x+1)^{15} (d-3 e)+\frac {5}{7} (x+1)^{14} (d-2 e)-\frac {5}{13} (x+1)^{13} (2 d-3 e)+\frac {1}{12} (x+1)^{12} (5 d-6 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{17} e (x+1)^{17}\) |
Input:
Int[x^5*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
-1/11*((d - e)*(1 + x)^11) + ((5*d - 6*e)*(1 + x)^12)/12 - (5*(2*d - 3*e)* (1 + x)^13)/13 + (5*(d - 2*e)*(1 + x)^14)/7 - ((d - 3*e)*(1 + x)^15)/3 + ( (d - 6*e)*(1 + x)^16)/16 + (e*(1 + x)^17)/17
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 = 0.89 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.25
| method | result | size |
| norman | \(\frac {d \,x^{6}}{6}+\left (\frac {10 d}{7}+\frac {e}{7}\right ) x^{7}+\left (\frac {45 d}{8}+\frac {5 e}{4}\right ) x^{8}+\left (\frac {40 d}{3}+5 e \right ) x^{9}+\left (21 d +12 e \right ) x^{10}+\left (\frac {252 d}{11}+\frac {210 e}{11}\right ) x^{11}+\left (\frac {35 d}{2}+21 e \right ) x^{12}+\left (\frac {120 d}{13}+\frac {210 e}{13}\right ) x^{13}+\left (\frac {45 d}{14}+\frac {60 e}{7}\right ) x^{14}+\left (\frac {2 d}{3}+3 e \right ) x^{15}+\left (\frac {d}{16}+\frac {5 e}{8}\right ) x^{16}+\frac {x^{17} e}{17}\) | \(124\) |
| default | \(\frac {x^{17} e}{17}+\frac {\left (d +10 e \right ) x^{16}}{16}+\frac {\left (10 d +45 e \right ) x^{15}}{15}+\frac {\left (45 d +120 e \right ) x^{14}}{14}+\frac {\left (120 d +210 e \right ) x^{13}}{13}+\frac {\left (210 d +252 e \right ) x^{12}}{12}+\frac {\left (252 d +210 e \right ) x^{11}}{11}+\frac {\left (210 d +120 e \right ) x^{10}}{10}+\frac {\left (120 d +45 e \right ) x^{9}}{9}+\frac {\left (45 d +10 e \right ) x^{8}}{8}+\frac {\left (10 d +e \right ) x^{7}}{7}+\frac {d \,x^{6}}{6}\) | \(130\) |
| gosper | \(\frac {x^{6} \left (48048 e \,x^{11}+51051 d \,x^{10}+510510 e \,x^{10}+544544 d \,x^{9}+2450448 e \,x^{9}+2625480 d \,x^{8}+7001280 e \,x^{8}+7539840 d \,x^{7}+13194720 e \,x^{7}+14294280 d \,x^{6}+17153136 e \,x^{6}+18712512 d \,x^{5}+15593760 x^{5} e +17153136 d \,x^{4}+9801792 x^{4} e +10890880 d \,x^{3}+4084080 x^{3} e +4594590 d \,x^{2}+1021020 e \,x^{2}+1166880 d x +116688 e x +136136 d \right )}{816816}\) | \(132\) |
| risch | \(\frac {1}{17} x^{17} e +\frac {1}{16} d \,x^{16}+\frac {5}{8} x^{16} e +\frac {2}{3} x^{15} d +3 x^{15} e +\frac {45}{14} x^{14} d +\frac {60}{7} x^{14} e +\frac {120}{13} d \,x^{13}+\frac {210}{13} x^{13} e +\frac {35}{2} d \,x^{12}+21 x^{12} e +\frac {252}{11} x^{11} d +\frac {210}{11} e \,x^{11}+21 d \,x^{10}+12 e \,x^{10}+\frac {40}{3} d \,x^{9}+5 e \,x^{9}+\frac {45}{8} d \,x^{8}+\frac {5}{4} e \,x^{8}+\frac {10}{7} d \,x^{7}+\frac {1}{7} e \,x^{7}+\frac {1}{6} d \,x^{6}\) | \(134\) |
| parallelrisch | \(\frac {1}{17} x^{17} e +\frac {1}{16} d \,x^{16}+\frac {5}{8} x^{16} e +\frac {2}{3} x^{15} d +3 x^{15} e +\frac {45}{14} x^{14} d +\frac {60}{7} x^{14} e +\frac {120}{13} d \,x^{13}+\frac {210}{13} x^{13} e +\frac {35}{2} d \,x^{12}+21 x^{12} e +\frac {252}{11} x^{11} d +\frac {210}{11} e \,x^{11}+21 d \,x^{10}+12 e \,x^{10}+\frac {40}{3} d \,x^{9}+5 e \,x^{9}+\frac {45}{8} d \,x^{8}+\frac {5}{4} e \,x^{8}+\frac {10}{7} d \,x^{7}+\frac {1}{7} e \,x^{7}+\frac {1}{6} d \,x^{6}\) | \(134\) |
| orering | \(\frac {x^{6} \left (48048 e \,x^{11}+51051 d \,x^{10}+510510 e \,x^{10}+544544 d \,x^{9}+2450448 e \,x^{9}+2625480 d \,x^{8}+7001280 e \,x^{8}+7539840 d \,x^{7}+13194720 e \,x^{7}+14294280 d \,x^{6}+17153136 e \,x^{6}+18712512 d \,x^{5}+15593760 x^{5} e +17153136 d \,x^{4}+9801792 x^{4} e +10890880 d \,x^{3}+4084080 x^{3} e +4594590 d \,x^{2}+1021020 e \,x^{2}+1166880 d x +116688 e x +136136 d \right ) \left (x^{2}+2 x +1\right )^{5}}{816816 \left (x +1\right )^{10}}\) | \(147\) |
Input:
int(x^5*(e*x+d)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/6*d*x^6+(10/7*d+1/7*e)*x^7+(45/8*d+5/4*e)*x^8+(40/3*d+5*e)*x^9+(21*d+12* e)*x^10+(252/11*d+210/11*e)*x^11+(35/2*d+21*e)*x^12+(120/13*d+210/13*e)*x^ 13+(45/14*d+60/7*e)*x^14+(2/3*d+3*e)*x^15+(1/16*d+5/8*e)*x^16+1/17*x^17*e
Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.30 \[ \int x^5 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{17} \, e x^{17} + \frac {1}{16} \, {\left (d + 10 \, e\right )} x^{16} + \frac {1}{3} \, {\left (2 \, d + 9 \, e\right )} x^{15} + \frac {15}{14} \, {\left (3 \, d + 8 \, e\right )} x^{14} + \frac {30}{13} \, {\left (4 \, d + 7 \, e\right )} x^{13} + \frac {7}{2} \, {\left (5 \, d + 6 \, e\right )} x^{12} + \frac {42}{11} \, {\left (6 \, d + 5 \, e\right )} x^{11} + 3 \, {\left (7 \, d + 4 \, e\right )} x^{10} + \frac {5}{3} \, {\left (8 \, d + 3 \, e\right )} x^{9} + \frac {5}{8} \, {\left (9 \, d + 2 \, e\right )} x^{8} + \frac {1}{7} \, {\left (10 \, d + e\right )} x^{7} + \frac {1}{6} \, d x^{6} \] Input:
integrate(x^5*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/17*e*x^17 + 1/16*(d + 10*e)*x^16 + 1/3*(2*d + 9*e)*x^15 + 15/14*(3*d + 8 *e)*x^14 + 30/13*(4*d + 7*e)*x^13 + 7/2*(5*d + 6*e)*x^12 + 42/11*(6*d + 5* e)*x^11 + 3*(7*d + 4*e)*x^10 + 5/3*(8*d + 3*e)*x^9 + 5/8*(9*d + 2*e)*x^8 + 1/7*(10*d + e)*x^7 + 1/6*d*x^6
Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.37 \[ \int x^5 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^{6}}{6} + \frac {e x^{17}}{17} + x^{16} \left (\frac {d}{16} + \frac {5 e}{8}\right ) + x^{15} \cdot \left (\frac {2 d}{3} + 3 e\right ) + x^{14} \cdot \left (\frac {45 d}{14} + \frac {60 e}{7}\right ) + x^{13} \cdot \left (\frac {120 d}{13} + \frac {210 e}{13}\right ) + x^{12} \cdot \left (\frac {35 d}{2} + 21 e\right ) + x^{11} \cdot \left (\frac {252 d}{11} + \frac {210 e}{11}\right ) + x^{10} \cdot \left (21 d + 12 e\right ) + x^{9} \cdot \left (\frac {40 d}{3} + 5 e\right ) + x^{8} \cdot \left (\frac {45 d}{8} + \frac {5 e}{4}\right ) + x^{7} \cdot \left (\frac {10 d}{7} + \frac {e}{7}\right ) \] Input:
integrate(x**5*(e*x+d)*(x**2+2*x+1)**5,x)
Output:
d*x**6/6 + e*x**17/17 + x**16*(d/16 + 5*e/8) + x**15*(2*d/3 + 3*e) + x**14 *(45*d/14 + 60*e/7) + x**13*(120*d/13 + 210*e/13) + x**12*(35*d/2 + 21*e) + x**11*(252*d/11 + 210*e/11) + x**10*(21*d + 12*e) + x**9*(40*d/3 + 5*e) + x**8*(45*d/8 + 5*e/4) + x**7*(10*d/7 + e/7)
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.30 \[ \int x^5 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{17} \, e x^{17} + \frac {1}{16} \, {\left (d + 10 \, e\right )} x^{16} + \frac {1}{3} \, {\left (2 \, d + 9 \, e\right )} x^{15} + \frac {15}{14} \, {\left (3 \, d + 8 \, e\right )} x^{14} + \frac {30}{13} \, {\left (4 \, d + 7 \, e\right )} x^{13} + \frac {7}{2} \, {\left (5 \, d + 6 \, e\right )} x^{12} + \frac {42}{11} \, {\left (6 \, d + 5 \, e\right )} x^{11} + 3 \, {\left (7 \, d + 4 \, e\right )} x^{10} + \frac {5}{3} \, {\left (8 \, d + 3 \, e\right )} x^{9} + \frac {5}{8} \, {\left (9 \, d + 2 \, e\right )} x^{8} + \frac {1}{7} \, {\left (10 \, d + e\right )} x^{7} + \frac {1}{6} \, d x^{6} \] Input:
integrate(x^5*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/17*e*x^17 + 1/16*(d + 10*e)*x^16 + 1/3*(2*d + 9*e)*x^15 + 15/14*(3*d + 8 *e)*x^14 + 30/13*(4*d + 7*e)*x^13 + 7/2*(5*d + 6*e)*x^12 + 42/11*(6*d + 5* e)*x^11 + 3*(7*d + 4*e)*x^10 + 5/3*(8*d + 3*e)*x^9 + 5/8*(9*d + 2*e)*x^8 + 1/7*(10*d + e)*x^7 + 1/6*d*x^6
Time = 0.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.34 \[ \int x^5 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{17} \, e x^{17} + \frac {1}{16} \, d x^{16} + \frac {5}{8} \, e x^{16} + \frac {2}{3} \, d x^{15} + 3 \, e x^{15} + \frac {45}{14} \, d x^{14} + \frac {60}{7} \, e x^{14} + \frac {120}{13} \, d x^{13} + \frac {210}{13} \, e x^{13} + \frac {35}{2} \, d x^{12} + 21 \, e x^{12} + \frac {252}{11} \, d x^{11} + \frac {210}{11} \, e x^{11} + 21 \, d x^{10} + 12 \, e x^{10} + \frac {40}{3} \, d x^{9} + 5 \, e x^{9} + \frac {45}{8} \, d x^{8} + \frac {5}{4} \, e x^{8} + \frac {10}{7} \, d x^{7} + \frac {1}{7} \, e x^{7} + \frac {1}{6} \, d x^{6} \] Input:
integrate(x^5*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/17*e*x^17 + 1/16*d*x^16 + 5/8*e*x^16 + 2/3*d*x^15 + 3*e*x^15 + 45/14*d*x ^14 + 60/7*e*x^14 + 120/13*d*x^13 + 210/13*e*x^13 + 35/2*d*x^12 + 21*e*x^1 2 + 252/11*d*x^11 + 210/11*e*x^11 + 21*d*x^10 + 12*e*x^10 + 40/3*d*x^9 + 5 *e*x^9 + 45/8*d*x^8 + 5/4*e*x^8 + 10/7*d*x^7 + 1/7*e*x^7 + 1/6*d*x^6
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.24 \[ \int x^5 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {e\,x^{17}}{17}+\left (\frac {d}{16}+\frac {5\,e}{8}\right )\,x^{16}+\left (\frac {2\,d}{3}+3\,e\right )\,x^{15}+\left (\frac {45\,d}{14}+\frac {60\,e}{7}\right )\,x^{14}+\left (\frac {120\,d}{13}+\frac {210\,e}{13}\right )\,x^{13}+\left (\frac {35\,d}{2}+21\,e\right )\,x^{12}+\left (\frac {252\,d}{11}+\frac {210\,e}{11}\right )\,x^{11}+\left (21\,d+12\,e\right )\,x^{10}+\left (\frac {40\,d}{3}+5\,e\right )\,x^9+\left (\frac {45\,d}{8}+\frac {5\,e}{4}\right )\,x^8+\left (\frac {10\,d}{7}+\frac {e}{7}\right )\,x^7+\frac {d\,x^6}{6} \] Input:
int(x^5*(d + e*x)*(2*x + x^2 + 1)^5,x)
Output:
x^15*((2*d)/3 + 3*e) + x^7*((10*d)/7 + e/7) + x^10*(21*d + 12*e) + x^16*(d /16 + (5*e)/8) + x^9*((40*d)/3 + 5*e) + x^8*((45*d)/8 + (5*e)/4) + x^12*(( 35*d)/2 + 21*e) + x^14*((45*d)/14 + (60*e)/7) + x^13*((120*d)/13 + (210*e) /13) + x^11*((252*d)/11 + (210*e)/11) + (d*x^6)/6 + (e*x^17)/17
Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.32 \[ \int x^5 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{6} \left (48048 e \,x^{11}+51051 d \,x^{10}+510510 e \,x^{10}+544544 d \,x^{9}+2450448 e \,x^{9}+2625480 d \,x^{8}+7001280 e \,x^{8}+7539840 d \,x^{7}+13194720 e \,x^{7}+14294280 d \,x^{6}+17153136 e \,x^{6}+18712512 d \,x^{5}+15593760 e \,x^{5}+17153136 d \,x^{4}+9801792 e \,x^{4}+10890880 d \,x^{3}+4084080 e \,x^{3}+4594590 d \,x^{2}+1021020 e \,x^{2}+1166880 d x +116688 e x +136136 d \right )}{816816} \] Input:
int(x^5*(e*x+d)*(x^2+2*x+1)^5,x)
Output:
(x**6*(51051*d*x**10 + 544544*d*x**9 + 2625480*d*x**8 + 7539840*d*x**7 + 1 4294280*d*x**6 + 18712512*d*x**5 + 17153136*d*x**4 + 10890880*d*x**3 + 459 4590*d*x**2 + 1166880*d*x + 136136*d + 48048*e*x**11 + 510510*e*x**10 + 24 50448*e*x**9 + 7001280*e*x**8 + 13194720*e*x**7 + 17153136*e*x**6 + 155937 60*e*x**5 + 9801792*e*x**4 + 4084080*e*x**3 + 1021020*e*x**2 + 116688*e*x) )/816816