Integrand size = 19, antiderivative size = 87 \[ \int x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{11} (d-e) (1+x)^{11}-\frac {1}{12} (4 d-5 e) (1+x)^{12}+\frac {2}{13} (3 d-5 e) (1+x)^{13}-\frac {1}{7} (2 d-5 e) (1+x)^{14}+\frac {1}{15} (d-5 e) (1+x)^{15}+\frac {1}{16} e (1+x)^{16} \] Output:
1/11*(d-e)*(1+x)^11-1/12*(4*d-5*e)*(1+x)^12+2/13*(3*d-5*e)*(1+x)^13-1/7*(2 *d-5*e)*(1+x)^14+1/15*(d-5*e)*(1+x)^15+1/16*e*(1+x)^16
Time = 0.02 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.76 \[ \int x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^5}{5}+\frac {1}{6} (10 d+e) x^6+\frac {5}{7} (9 d+2 e) x^7+\frac {15}{8} (8 d+3 e) x^8+\frac {10}{3} (7 d+4 e) x^9+\frac {21}{5} (6 d+5 e) x^{10}+\frac {42}{11} (5 d+6 e) x^{11}+\frac {5}{2} (4 d+7 e) x^{12}+\frac {15}{13} (3 d+8 e) x^{13}+\frac {5}{14} (2 d+9 e) x^{14}+\frac {1}{15} (d+10 e) x^{15}+\frac {e x^{16}}{16} \] Input:
Integrate[x^4*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
(d*x^5)/5 + ((10*d + e)*x^6)/6 + (5*(9*d + 2*e)*x^7)/7 + (15*(8*d + 3*e)*x ^8)/8 + (10*(7*d + 4*e)*x^9)/3 + (21*(6*d + 5*e)*x^10)/5 + (42*(5*d + 6*e) *x^11)/11 + (5*(4*d + 7*e)*x^12)/2 + (15*(3*d + 8*e)*x^13)/13 + (5*(2*d + 9*e)*x^14)/14 + ((d + 10*e)*x^15)/15 + (e*x^16)/16
Time = 0.46 (sec) , antiderivative size = 87, 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.
| \(\displaystyle \int x^4 \left (x^2+2 x+1\right )^5 (d+e x) \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \int x^4 (x+1)^{10} (d+e x)dx\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \int \left ((x+1)^{14} (d-5 e)-2 (x+1)^{13} (2 d-5 e)+2 (x+1)^{12} (3 d-5 e)+(x+1)^{11} (5 e-4 d)+(x+1)^{10} (d-e)+e (x+1)^{15}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{15} (x+1)^{15} (d-5 e)-\frac {1}{7} (x+1)^{14} (2 d-5 e)+\frac {2}{13} (x+1)^{13} (3 d-5 e)-\frac {1}{12} (x+1)^{12} (4 d-5 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{16} e (x+1)^{16}\) |
Input:
Int[x^4*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
((d - e)*(1 + x)^11)/11 - ((4*d - 5*e)*(1 + x)^12)/12 + (2*(3*d - 5*e)*(1 + x)^13)/13 - ((2*d - 5*e)*(1 + x)^14)/7 + ((d - 5*e)*(1 + x)^15)/15 + (e* (1 + x)^16)/16
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 = 0.81 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.43
| method | result | size |
| norman | \(\frac {d \,x^{5}}{5}+\left (\frac {5 d}{3}+\frac {e}{6}\right ) x^{6}+\left (\frac {45 d}{7}+\frac {10 e}{7}\right ) x^{7}+\left (15 d +\frac {45 e}{8}\right ) x^{8}+\left (\frac {70 d}{3}+\frac {40 e}{3}\right ) x^{9}+\left (\frac {126 d}{5}+21 e \right ) x^{10}+\left (\frac {210 d}{11}+\frac {252 e}{11}\right ) x^{11}+\left (10 d +\frac {35 e}{2}\right ) x^{12}+\left (\frac {45 d}{13}+\frac {120 e}{13}\right ) x^{13}+\left (\frac {5 d}{7}+\frac {45 e}{14}\right ) x^{14}+\left (\frac {d}{15}+\frac {2 e}{3}\right ) x^{15}+\frac {x^{16} e}{16}\) | \(124\) |
| default | \(\frac {x^{16} e}{16}+\frac {\left (d +10 e \right ) x^{15}}{15}+\frac {\left (10 d +45 e \right ) x^{14}}{14}+\frac {\left (45 d +120 e \right ) x^{13}}{13}+\frac {\left (120 d +210 e \right ) x^{12}}{12}+\frac {\left (210 d +252 e \right ) x^{11}}{11}+\frac {\left (252 d +210 e \right ) x^{10}}{10}+\frac {\left (210 d +120 e \right ) x^{9}}{9}+\frac {\left (120 d +45 e \right ) x^{8}}{8}+\frac {\left (45 d +10 e \right ) x^{7}}{7}+\frac {\left (10 d +e \right ) x^{6}}{6}+\frac {d \,x^{5}}{5}\) | \(130\) |
| gosper | \(\frac {x^{5} \left (15015 e \,x^{11}+16016 d \,x^{10}+160160 e \,x^{10}+171600 d \,x^{9}+772200 e \,x^{9}+831600 d \,x^{8}+2217600 e \,x^{8}+2402400 d \,x^{7}+4204200 e \,x^{7}+4586400 d \,x^{6}+5503680 e \,x^{6}+6054048 d \,x^{5}+5045040 x^{5} e +5605600 d \,x^{4}+3203200 x^{4} e +3603600 d \,x^{3}+1351350 x^{3} e +1544400 d \,x^{2}+343200 e \,x^{2}+400400 d x +40040 e x +48048 d \right )}{240240}\) | \(132\) |
| risch | \(\frac {1}{16} x^{16} e +\frac {1}{15} x^{15} d +\frac {2}{3} x^{15} e +\frac {5}{7} x^{14} d +\frac {45}{14} x^{14} e +\frac {45}{13} d \,x^{13}+\frac {120}{13} x^{13} e +10 d \,x^{12}+\frac {35}{2} x^{12} e +\frac {210}{11} x^{11} d +\frac {252}{11} e \,x^{11}+\frac {126}{5} d \,x^{10}+21 e \,x^{10}+\frac {70}{3} d \,x^{9}+\frac {40}{3} e \,x^{9}+15 d \,x^{8}+\frac {45}{8} e \,x^{8}+\frac {45}{7} d \,x^{7}+\frac {10}{7} e \,x^{7}+\frac {5}{3} d \,x^{6}+\frac {1}{6} e \,x^{6}+\frac {1}{5} d \,x^{5}\) | \(134\) |
| parallelrisch | \(\frac {1}{16} x^{16} e +\frac {1}{15} x^{15} d +\frac {2}{3} x^{15} e +\frac {5}{7} x^{14} d +\frac {45}{14} x^{14} e +\frac {45}{13} d \,x^{13}+\frac {120}{13} x^{13} e +10 d \,x^{12}+\frac {35}{2} x^{12} e +\frac {210}{11} x^{11} d +\frac {252}{11} e \,x^{11}+\frac {126}{5} d \,x^{10}+21 e \,x^{10}+\frac {70}{3} d \,x^{9}+\frac {40}{3} e \,x^{9}+15 d \,x^{8}+\frac {45}{8} e \,x^{8}+\frac {45}{7} d \,x^{7}+\frac {10}{7} e \,x^{7}+\frac {5}{3} d \,x^{6}+\frac {1}{6} e \,x^{6}+\frac {1}{5} d \,x^{5}\) | \(134\) |
| orering | \(\frac {x^{5} \left (15015 e \,x^{11}+16016 d \,x^{10}+160160 e \,x^{10}+171600 d \,x^{9}+772200 e \,x^{9}+831600 d \,x^{8}+2217600 e \,x^{8}+2402400 d \,x^{7}+4204200 e \,x^{7}+4586400 d \,x^{6}+5503680 e \,x^{6}+6054048 d \,x^{5}+5045040 x^{5} e +5605600 d \,x^{4}+3203200 x^{4} e +3603600 d \,x^{3}+1351350 x^{3} e +1544400 d \,x^{2}+343200 e \,x^{2}+400400 d x +40040 e x +48048 d \right ) \left (x^{2}+2 x +1\right )^{5}}{240240 \left (x +1\right )^{10}}\) | \(147\) |
Input:
int(x^4*(e*x+d)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/5*d*x^5+(5/3*d+1/6*e)*x^6+(45/7*d+10/7*e)*x^7+(15*d+45/8*e)*x^8+(70/3*d+ 40/3*e)*x^9+(126/5*d+21*e)*x^10+(210/11*d+252/11*e)*x^11+(10*d+35/2*e)*x^1 2+(45/13*d+120/13*e)*x^13+(5/7*d+45/14*e)*x^14+(1/15*d+2/3*e)*x^15+1/16*x^ 16*e
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.48 \[ \int x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{16} \, e x^{16} + \frac {1}{15} \, {\left (d + 10 \, e\right )} x^{15} + \frac {5}{14} \, {\left (2 \, d + 9 \, e\right )} x^{14} + \frac {15}{13} \, {\left (3 \, d + 8 \, e\right )} x^{13} + \frac {5}{2} \, {\left (4 \, d + 7 \, e\right )} x^{12} + \frac {42}{11} \, {\left (5 \, d + 6 \, e\right )} x^{11} + \frac {21}{5} \, {\left (6 \, d + 5 \, e\right )} x^{10} + \frac {10}{3} \, {\left (7 \, d + 4 \, e\right )} x^{9} + \frac {15}{8} \, {\left (8 \, d + 3 \, e\right )} x^{8} + \frac {5}{7} \, {\left (9 \, d + 2 \, e\right )} x^{7} + \frac {1}{6} \, {\left (10 \, d + e\right )} x^{6} + \frac {1}{5} \, d x^{5} \] Input:
integrate(x^4*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/16*e*x^16 + 1/15*(d + 10*e)*x^15 + 5/14*(2*d + 9*e)*x^14 + 15/13*(3*d + 8*e)*x^13 + 5/2*(4*d + 7*e)*x^12 + 42/11*(5*d + 6*e)*x^11 + 21/5*(6*d + 5* e)*x^10 + 10/3*(7*d + 4*e)*x^9 + 15/8*(8*d + 3*e)*x^8 + 5/7*(9*d + 2*e)*x^ 7 + 1/6*(10*d + e)*x^6 + 1/5*d*x^5
Time = 0.04 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.60 \[ \int x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^{5}}{5} + \frac {e x^{16}}{16} + x^{15} \left (\frac {d}{15} + \frac {2 e}{3}\right ) + x^{14} \cdot \left (\frac {5 d}{7} + \frac {45 e}{14}\right ) + x^{13} \cdot \left (\frac {45 d}{13} + \frac {120 e}{13}\right ) + x^{12} \cdot \left (10 d + \frac {35 e}{2}\right ) + x^{11} \cdot \left (\frac {210 d}{11} + \frac {252 e}{11}\right ) + x^{10} \cdot \left (\frac {126 d}{5} + 21 e\right ) + x^{9} \cdot \left (\frac {70 d}{3} + \frac {40 e}{3}\right ) + x^{8} \cdot \left (15 d + \frac {45 e}{8}\right ) + x^{7} \cdot \left (\frac {45 d}{7} + \frac {10 e}{7}\right ) + x^{6} \cdot \left (\frac {5 d}{3} + \frac {e}{6}\right ) \] Input:
integrate(x**4*(e*x+d)*(x**2+2*x+1)**5,x)
Output:
d*x**5/5 + e*x**16/16 + x**15*(d/15 + 2*e/3) + x**14*(5*d/7 + 45*e/14) + x **13*(45*d/13 + 120*e/13) + x**12*(10*d + 35*e/2) + x**11*(210*d/11 + 252* e/11) + x**10*(126*d/5 + 21*e) + x**9*(70*d/3 + 40*e/3) + x**8*(15*d + 45* e/8) + x**7*(45*d/7 + 10*e/7) + x**6*(5*d/3 + e/6)
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.48 \[ \int x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{16} \, e x^{16} + \frac {1}{15} \, {\left (d + 10 \, e\right )} x^{15} + \frac {5}{14} \, {\left (2 \, d + 9 \, e\right )} x^{14} + \frac {15}{13} \, {\left (3 \, d + 8 \, e\right )} x^{13} + \frac {5}{2} \, {\left (4 \, d + 7 \, e\right )} x^{12} + \frac {42}{11} \, {\left (5 \, d + 6 \, e\right )} x^{11} + \frac {21}{5} \, {\left (6 \, d + 5 \, e\right )} x^{10} + \frac {10}{3} \, {\left (7 \, d + 4 \, e\right )} x^{9} + \frac {15}{8} \, {\left (8 \, d + 3 \, e\right )} x^{8} + \frac {5}{7} \, {\left (9 \, d + 2 \, e\right )} x^{7} + \frac {1}{6} \, {\left (10 \, d + e\right )} x^{6} + \frac {1}{5} \, d x^{5} \] Input:
integrate(x^4*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/16*e*x^16 + 1/15*(d + 10*e)*x^15 + 5/14*(2*d + 9*e)*x^14 + 15/13*(3*d + 8*e)*x^13 + 5/2*(4*d + 7*e)*x^12 + 42/11*(5*d + 6*e)*x^11 + 21/5*(6*d + 5* e)*x^10 + 10/3*(7*d + 4*e)*x^9 + 15/8*(8*d + 3*e)*x^8 + 5/7*(9*d + 2*e)*x^ 7 + 1/6*(10*d + e)*x^6 + 1/5*d*x^5
Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.53 \[ \int x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{16} \, e x^{16} + \frac {1}{15} \, d x^{15} + \frac {2}{3} \, e x^{15} + \frac {5}{7} \, d x^{14} + \frac {45}{14} \, e x^{14} + \frac {45}{13} \, d x^{13} + \frac {120}{13} \, e x^{13} + 10 \, d x^{12} + \frac {35}{2} \, e x^{12} + \frac {210}{11} \, d x^{11} + \frac {252}{11} \, e x^{11} + \frac {126}{5} \, d x^{10} + 21 \, e x^{10} + \frac {70}{3} \, d x^{9} + \frac {40}{3} \, e x^{9} + 15 \, d x^{8} + \frac {45}{8} \, e x^{8} + \frac {45}{7} \, d x^{7} + \frac {10}{7} \, e x^{7} + \frac {5}{3} \, d x^{6} + \frac {1}{6} \, e x^{6} + \frac {1}{5} \, d x^{5} \] Input:
integrate(x^4*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/16*e*x^16 + 1/15*d*x^15 + 2/3*e*x^15 + 5/7*d*x^14 + 45/14*e*x^14 + 45/13 *d*x^13 + 120/13*e*x^13 + 10*d*x^12 + 35/2*e*x^12 + 210/11*d*x^11 + 252/11 *e*x^11 + 126/5*d*x^10 + 21*e*x^10 + 70/3*d*x^9 + 40/3*e*x^9 + 15*d*x^8 + 45/8*e*x^8 + 45/7*d*x^7 + 10/7*e*x^7 + 5/3*d*x^6 + 1/6*e*x^6 + 1/5*d*x^5
Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.41 \[ \int x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {e\,x^{16}}{16}+\left (\frac {d}{15}+\frac {2\,e}{3}\right )\,x^{15}+\left (\frac {5\,d}{7}+\frac {45\,e}{14}\right )\,x^{14}+\left (\frac {45\,d}{13}+\frac {120\,e}{13}\right )\,x^{13}+\left (10\,d+\frac {35\,e}{2}\right )\,x^{12}+\left (\frac {210\,d}{11}+\frac {252\,e}{11}\right )\,x^{11}+\left (\frac {126\,d}{5}+21\,e\right )\,x^{10}+\left (\frac {70\,d}{3}+\frac {40\,e}{3}\right )\,x^9+\left (15\,d+\frac {45\,e}{8}\right )\,x^8+\left (\frac {45\,d}{7}+\frac {10\,e}{7}\right )\,x^7+\left (\frac {5\,d}{3}+\frac {e}{6}\right )\,x^6+\frac {d\,x^5}{5} \] Input:
int(x^4*(d + e*x)*(2*x + x^2 + 1)^5,x)
Output:
x^6*((5*d)/3 + e/6) + x^15*(d/15 + (2*e)/3) + x^12*(10*d + (35*e)/2) + x^7 *((45*d)/7 + (10*e)/7) + x^8*(15*d + (45*e)/8) + x^14*((5*d)/7 + (45*e)/14 ) + x^9*((70*d)/3 + (40*e)/3) + x^10*((126*d)/5 + 21*e) + x^13*((45*d)/13 + (120*e)/13) + x^11*((210*d)/11 + (252*e)/11) + (d*x^5)/5 + (e*x^16)/16
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.51 \[ \int x^4 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{5} \left (15015 e \,x^{11}+16016 d \,x^{10}+160160 e \,x^{10}+171600 d \,x^{9}+772200 e \,x^{9}+831600 d \,x^{8}+2217600 e \,x^{8}+2402400 d \,x^{7}+4204200 e \,x^{7}+4586400 d \,x^{6}+5503680 e \,x^{6}+6054048 d \,x^{5}+5045040 e \,x^{5}+5605600 d \,x^{4}+3203200 e \,x^{4}+3603600 d \,x^{3}+1351350 e \,x^{3}+1544400 d \,x^{2}+343200 e \,x^{2}+400400 d x +40040 e x +48048 d \right )}{240240} \] Input:
int(x^4*(e*x+d)*(x^2+2*x+1)^5,x)
Output:
(x**5*(16016*d*x**10 + 171600*d*x**9 + 831600*d*x**8 + 2402400*d*x**7 + 45 86400*d*x**6 + 6054048*d*x**5 + 5605600*d*x**4 + 3603600*d*x**3 + 1544400* d*x**2 + 400400*d*x + 48048*d + 15015*e*x**11 + 160160*e*x**10 + 772200*e* x**9 + 2217600*e*x**8 + 4204200*e*x**7 + 5503680*e*x**6 + 5045040*e*x**5 + 3203200*e*x**4 + 1351350*e*x**3 + 343200*e*x**2 + 40040*e*x))/240240