Integrand size = 19, antiderivative size = 119 \[ \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{11} (d-e) (1+x)^{11}-\frac {1}{12} (6 d-7 e) (1+x)^{12}+\frac {3}{13} (5 d-7 e) (1+x)^{13}-\frac {5}{14} (4 d-7 e) (1+x)^{14}+\frac {1}{3} (3 d-7 e) (1+x)^{15}-\frac {3}{16} (2 d-7 e) (1+x)^{16}+\frac {1}{17} (d-7 e) (1+x)^{17}+\frac {1}{18} e (1+x)^{18} \] Output:
1/11*(d-e)*(1+x)^11-1/12*(6*d-7*e)*(1+x)^12+3/13*(5*d-7*e)*(1+x)^13-5/14*( 4*d-7*e)*(1+x)^14+1/3*(3*d-7*e)*(1+x)^15-3/16*(2*d-7*e)*(1+x)^16+1/17*(d-7 *e)*(1+x)^17+1/18*e*(1+x)^18
Time = 0.02 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.26 \[ \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^7}{7}+\frac {1}{8} (10 d+e) x^8+\frac {5}{9} (9 d+2 e) x^9+\frac {3}{2} (8 d+3 e) x^{10}+\frac {30}{11} (7 d+4 e) x^{11}+\frac {7}{2} (6 d+5 e) x^{12}+\frac {42}{13} (5 d+6 e) x^{13}+\frac {15}{7} (4 d+7 e) x^{14}+(3 d+8 e) x^{15}+\frac {5}{16} (2 d+9 e) x^{16}+\frac {1}{17} (d+10 e) x^{17}+\frac {e x^{18}}{18} \] Input:
Integrate[x^6*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
(d*x^7)/7 + ((10*d + e)*x^8)/8 + (5*(9*d + 2*e)*x^9)/9 + (3*(8*d + 3*e)*x^ 10)/2 + (30*(7*d + 4*e)*x^11)/11 + (7*(6*d + 5*e)*x^12)/2 + (42*(5*d + 6*e )*x^13)/13 + (15*(4*d + 7*e)*x^14)/7 + (3*d + 8*e)*x^15 + (5*(2*d + 9*e)*x ^16)/16 + ((d + 10*e)*x^17)/17 + (e*x^18)/18
Time = 0.50 (sec) , antiderivative size = 119, 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.
| \(\displaystyle \int x^6 \left (x^2+2 x+1\right )^5 (d+e x) \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \int x^6 (x+1)^{10} (d+e x)dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left ((x+1)^{16} (d-7 e)-3 (x+1)^{15} (2 d-7 e)+5 (x+1)^{14} (3 d-7 e)-5 (x+1)^{13} (4 d-7 e)+3 (x+1)^{12} (5 d-7 e)+(x+1)^{11} (7 e-6 d)+(x+1)^{10} (d-e)+e (x+1)^{17}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{17} (x+1)^{17} (d-7 e)-\frac {3}{16} (x+1)^{16} (2 d-7 e)+\frac {1}{3} (x+1)^{15} (3 d-7 e)-\frac {5}{14} (x+1)^{14} (4 d-7 e)+\frac {3}{13} (x+1)^{13} (5 d-7 e)-\frac {1}{12} (x+1)^{12} (6 d-7 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{18} e (x+1)^{18}\) |
Input:
Int[x^6*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
((d - e)*(1 + x)^11)/11 - ((6*d - 7*e)*(1 + x)^12)/12 + (3*(5*d - 7*e)*(1 + x)^13)/13 - (5*(4*d - 7*e)*(1 + x)^14)/14 + ((3*d - 7*e)*(1 + x)^15)/3 - (3*(2*d - 7*e)*(1 + x)^16)/16 + ((d - 7*e)*(1 + x)^17)/17 + (e*(1 + x)^18 )/18
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 = 0.86 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.04
| method | result | size |
| norman | \(\frac {d \,x^{7}}{7}+\left (\frac {5 d}{4}+\frac {e}{8}\right ) x^{8}+\left (5 d +\frac {10 e}{9}\right ) x^{9}+\left (12 d +\frac {9 e}{2}\right ) x^{10}+\left (\frac {210 d}{11}+\frac {120 e}{11}\right ) x^{11}+\left (21 d +\frac {35 e}{2}\right ) x^{12}+\left (\frac {210 d}{13}+\frac {252 e}{13}\right ) x^{13}+\left (\frac {60 d}{7}+15 e \right ) x^{14}+\left (3 d +8 e \right ) x^{15}+\left (\frac {5 d}{8}+\frac {45 e}{16}\right ) x^{16}+\left (\frac {d}{17}+\frac {10 e}{17}\right ) x^{17}+\frac {e \,x^{18}}{18}\) | \(124\) |
| default | \(\frac {e \,x^{18}}{18}+\frac {\left (d +10 e \right ) x^{17}}{17}+\frac {\left (10 d +45 e \right ) x^{16}}{16}+\frac {\left (45 d +120 e \right ) x^{15}}{15}+\frac {\left (120 d +210 e \right ) x^{14}}{14}+\frac {\left (210 d +252 e \right ) x^{13}}{13}+\frac {\left (252 d +210 e \right ) x^{12}}{12}+\frac {\left (210 d +120 e \right ) x^{11}}{11}+\frac {\left (120 d +45 e \right ) x^{10}}{10}+\frac {\left (45 d +10 e \right ) x^{9}}{9}+\frac {\left (10 d +e \right ) x^{8}}{8}+\frac {d \,x^{7}}{7}\) | \(130\) |
| gosper | \(\frac {x^{7} \left (136136 e \,x^{11}+144144 d \,x^{10}+1441440 e \,x^{10}+1531530 d \,x^{9}+6891885 e \,x^{9}+7351344 d \,x^{8}+19603584 e \,x^{8}+21003840 d \,x^{7}+36756720 e \,x^{7}+39584160 d \,x^{6}+47500992 e \,x^{6}+51459408 d \,x^{5}+42882840 x^{5} e +46781280 d \,x^{4}+26732160 x^{4} e +29405376 d \,x^{3}+11027016 x^{3} e +12252240 d \,x^{2}+2722720 e \,x^{2}+3063060 d x +306306 e x +350064 d \right )}{2450448}\) | \(132\) |
| risch | \(\frac {1}{18} e \,x^{18}+\frac {1}{17} d \,x^{17}+\frac {10}{17} x^{17} e +\frac {5}{8} d \,x^{16}+\frac {45}{16} x^{16} e +3 x^{15} d +8 x^{15} e +\frac {60}{7} x^{14} d +15 x^{14} e +\frac {210}{13} d \,x^{13}+\frac {252}{13} x^{13} e +21 d \,x^{12}+\frac {35}{2} x^{12} e +\frac {210}{11} x^{11} d +\frac {120}{11} e \,x^{11}+12 d \,x^{10}+\frac {9}{2} e \,x^{10}+5 d \,x^{9}+\frac {10}{9} e \,x^{9}+\frac {5}{4} d \,x^{8}+\frac {1}{8} e \,x^{8}+\frac {1}{7} d \,x^{7}\) | \(134\) |
| parallelrisch | \(\frac {1}{18} e \,x^{18}+\frac {1}{17} d \,x^{17}+\frac {10}{17} x^{17} e +\frac {5}{8} d \,x^{16}+\frac {45}{16} x^{16} e +3 x^{15} d +8 x^{15} e +\frac {60}{7} x^{14} d +15 x^{14} e +\frac {210}{13} d \,x^{13}+\frac {252}{13} x^{13} e +21 d \,x^{12}+\frac {35}{2} x^{12} e +\frac {210}{11} x^{11} d +\frac {120}{11} e \,x^{11}+12 d \,x^{10}+\frac {9}{2} e \,x^{10}+5 d \,x^{9}+\frac {10}{9} e \,x^{9}+\frac {5}{4} d \,x^{8}+\frac {1}{8} e \,x^{8}+\frac {1}{7} d \,x^{7}\) | \(134\) |
| orering | \(\frac {x^{7} \left (136136 e \,x^{11}+144144 d \,x^{10}+1441440 e \,x^{10}+1531530 d \,x^{9}+6891885 e \,x^{9}+7351344 d \,x^{8}+19603584 e \,x^{8}+21003840 d \,x^{7}+36756720 e \,x^{7}+39584160 d \,x^{6}+47500992 e \,x^{6}+51459408 d \,x^{5}+42882840 x^{5} e +46781280 d \,x^{4}+26732160 x^{4} e +29405376 d \,x^{3}+11027016 x^{3} e +12252240 d \,x^{2}+2722720 e \,x^{2}+3063060 d x +306306 e x +350064 d \right ) \left (x^{2}+2 x +1\right )^{5}}{2450448 \left (x +1\right )^{10}}\) | \(147\) |
Input:
int(x^6*(e*x+d)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/7*d*x^7+(5/4*d+1/8*e)*x^8+(5*d+10/9*e)*x^9+(12*d+9/2*e)*x^10+(210/11*d+1 20/11*e)*x^11+(21*d+35/2*e)*x^12+(210/13*d+252/13*e)*x^13+(60/7*d+15*e)*x^ 14+(3*d+8*e)*x^15+(5/8*d+45/16*e)*x^16+(1/17*d+10/17*e)*x^17+1/18*e*x^18
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08 \[ \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{18} \, e x^{18} + \frac {1}{17} \, {\left (d + 10 \, e\right )} x^{17} + \frac {5}{16} \, {\left (2 \, d + 9 \, e\right )} x^{16} + {\left (3 \, d + 8 \, e\right )} x^{15} + \frac {15}{7} \, {\left (4 \, d + 7 \, e\right )} x^{14} + \frac {42}{13} \, {\left (5 \, d + 6 \, e\right )} x^{13} + \frac {7}{2} \, {\left (6 \, d + 5 \, e\right )} x^{12} + \frac {30}{11} \, {\left (7 \, d + 4 \, e\right )} x^{11} + \frac {3}{2} \, {\left (8 \, d + 3 \, e\right )} x^{10} + \frac {5}{9} \, {\left (9 \, d + 2 \, e\right )} x^{9} + \frac {1}{8} \, {\left (10 \, d + e\right )} x^{8} + \frac {1}{7} \, d x^{7} \] Input:
integrate(x^6*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/18*e*x^18 + 1/17*(d + 10*e)*x^17 + 5/16*(2*d + 9*e)*x^16 + (3*d + 8*e)*x ^15 + 15/7*(4*d + 7*e)*x^14 + 42/13*(5*d + 6*e)*x^13 + 7/2*(6*d + 5*e)*x^1 2 + 30/11*(7*d + 4*e)*x^11 + 3/2*(8*d + 3*e)*x^10 + 5/9*(9*d + 2*e)*x^9 + 1/8*(10*d + e)*x^8 + 1/7*d*x^7
Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.13 \[ \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^{7}}{7} + \frac {e x^{18}}{18} + x^{17} \left (\frac {d}{17} + \frac {10 e}{17}\right ) + x^{16} \cdot \left (\frac {5 d}{8} + \frac {45 e}{16}\right ) + x^{15} \cdot \left (3 d + 8 e\right ) + x^{14} \cdot \left (\frac {60 d}{7} + 15 e\right ) + x^{13} \cdot \left (\frac {210 d}{13} + \frac {252 e}{13}\right ) + x^{12} \cdot \left (21 d + \frac {35 e}{2}\right ) + x^{11} \cdot \left (\frac {210 d}{11} + \frac {120 e}{11}\right ) + x^{10} \cdot \left (12 d + \frac {9 e}{2}\right ) + x^{9} \cdot \left (5 d + \frac {10 e}{9}\right ) + x^{8} \cdot \left (\frac {5 d}{4} + \frac {e}{8}\right ) \] Input:
integrate(x**6*(e*x+d)*(x**2+2*x+1)**5,x)
Output:
d*x**7/7 + e*x**18/18 + x**17*(d/17 + 10*e/17) + x**16*(5*d/8 + 45*e/16) + x**15*(3*d + 8*e) + x**14*(60*d/7 + 15*e) + x**13*(210*d/13 + 252*e/13) + x**12*(21*d + 35*e/2) + x**11*(210*d/11 + 120*e/11) + x**10*(12*d + 9*e/2 ) + x**9*(5*d + 10*e/9) + x**8*(5*d/4 + e/8)
Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08 \[ \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{18} \, e x^{18} + \frac {1}{17} \, {\left (d + 10 \, e\right )} x^{17} + \frac {5}{16} \, {\left (2 \, d + 9 \, e\right )} x^{16} + {\left (3 \, d + 8 \, e\right )} x^{15} + \frac {15}{7} \, {\left (4 \, d + 7 \, e\right )} x^{14} + \frac {42}{13} \, {\left (5 \, d + 6 \, e\right )} x^{13} + \frac {7}{2} \, {\left (6 \, d + 5 \, e\right )} x^{12} + \frac {30}{11} \, {\left (7 \, d + 4 \, e\right )} x^{11} + \frac {3}{2} \, {\left (8 \, d + 3 \, e\right )} x^{10} + \frac {5}{9} \, {\left (9 \, d + 2 \, e\right )} x^{9} + \frac {1}{8} \, {\left (10 \, d + e\right )} x^{8} + \frac {1}{7} \, d x^{7} \] Input:
integrate(x^6*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/18*e*x^18 + 1/17*(d + 10*e)*x^17 + 5/16*(2*d + 9*e)*x^16 + (3*d + 8*e)*x ^15 + 15/7*(4*d + 7*e)*x^14 + 42/13*(5*d + 6*e)*x^13 + 7/2*(6*d + 5*e)*x^1 2 + 30/11*(7*d + 4*e)*x^11 + 3/2*(8*d + 3*e)*x^10 + 5/9*(9*d + 2*e)*x^9 + 1/8*(10*d + e)*x^8 + 1/7*d*x^7
Time = 0.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.12 \[ \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{18} \, e x^{18} + \frac {1}{17} \, d x^{17} + \frac {10}{17} \, e x^{17} + \frac {5}{8} \, d x^{16} + \frac {45}{16} \, e x^{16} + 3 \, d x^{15} + 8 \, e x^{15} + \frac {60}{7} \, d x^{14} + 15 \, e x^{14} + \frac {210}{13} \, d x^{13} + \frac {252}{13} \, e x^{13} + 21 \, d x^{12} + \frac {35}{2} \, e x^{12} + \frac {210}{11} \, d x^{11} + \frac {120}{11} \, e x^{11} + 12 \, d x^{10} + \frac {9}{2} \, e x^{10} + 5 \, d x^{9} + \frac {10}{9} \, e x^{9} + \frac {5}{4} \, d x^{8} + \frac {1}{8} \, e x^{8} + \frac {1}{7} \, d x^{7} \] Input:
integrate(x^6*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/18*e*x^18 + 1/17*d*x^17 + 10/17*e*x^17 + 5/8*d*x^16 + 45/16*e*x^16 + 3*d *x^15 + 8*e*x^15 + 60/7*d*x^14 + 15*e*x^14 + 210/13*d*x^13 + 252/13*e*x^13 + 21*d*x^12 + 35/2*e*x^12 + 210/11*d*x^11 + 120/11*e*x^11 + 12*d*x^10 + 9 /2*e*x^10 + 5*d*x^9 + 10/9*e*x^9 + 5/4*d*x^8 + 1/8*e*x^8 + 1/7*d*x^7
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.03 \[ \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {e\,x^{18}}{18}+\left (\frac {d}{17}+\frac {10\,e}{17}\right )\,x^{17}+\left (\frac {5\,d}{8}+\frac {45\,e}{16}\right )\,x^{16}+\left (3\,d+8\,e\right )\,x^{15}+\left (\frac {60\,d}{7}+15\,e\right )\,x^{14}+\left (\frac {210\,d}{13}+\frac {252\,e}{13}\right )\,x^{13}+\left (21\,d+\frac {35\,e}{2}\right )\,x^{12}+\left (\frac {210\,d}{11}+\frac {120\,e}{11}\right )\,x^{11}+\left (12\,d+\frac {9\,e}{2}\right )\,x^{10}+\left (5\,d+\frac {10\,e}{9}\right )\,x^9+\left (\frac {5\,d}{4}+\frac {e}{8}\right )\,x^8+\frac {d\,x^7}{7} \] Input:
int(x^6*(d + e*x)*(2*x + x^2 + 1)^5,x)
Output:
x^8*((5*d)/4 + e/8) + x^15*(3*d + 8*e) + x^9*(5*d + (10*e)/9) + x^10*(12*d + (9*e)/2) + x^17*(d/17 + (10*e)/17) + x^12*(21*d + (35*e)/2) + x^16*((5* d)/8 + (45*e)/16) + x^14*((60*d)/7 + 15*e) + x^11*((210*d)/11 + (120*e)/11 ) + x^13*((210*d)/13 + (252*e)/13) + (d*x^7)/7 + (e*x^18)/18
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.10 \[ \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{7} \left (136136 e \,x^{11}+144144 d \,x^{10}+1441440 e \,x^{10}+1531530 d \,x^{9}+6891885 e \,x^{9}+7351344 d \,x^{8}+19603584 e \,x^{8}+21003840 d \,x^{7}+36756720 e \,x^{7}+39584160 d \,x^{6}+47500992 e \,x^{6}+51459408 d \,x^{5}+42882840 e \,x^{5}+46781280 d \,x^{4}+26732160 e \,x^{4}+29405376 d \,x^{3}+11027016 e \,x^{3}+12252240 d \,x^{2}+2722720 e \,x^{2}+3063060 d x +306306 e x +350064 d \right )}{2450448} \] Input:
int(x^6*(e*x+d)*(x^2+2*x+1)^5,x)
Output:
(x**7*(144144*d*x**10 + 1531530*d*x**9 + 7351344*d*x**8 + 21003840*d*x**7 + 39584160*d*x**6 + 51459408*d*x**5 + 46781280*d*x**4 + 29405376*d*x**3 + 12252240*d*x**2 + 3063060*d*x + 350064*d + 136136*e*x**11 + 1441440*e*x**1 0 + 6891885*e*x**9 + 19603584*e*x**8 + 36756720*e*x**7 + 47500992*e*x**6 + 42882840*e*x**5 + 26732160*e*x**4 + 11027016*e*x**3 + 2722720*e*x**2 + 30 6306*e*x))/2450448