3.51 \(\int (1-x)^{20} x^4 \, dx\)

Optimal. Leaf size=56 \[ -\frac{1}{25} (1-x)^{25}+\frac{1}{6} (1-x)^{24}-\frac{6}{23} (1-x)^{23}+\frac{2}{11} (1-x)^{22}-\frac{1}{21} (1-x)^{21} \]

[Out]

-(1 - x)^21/21 + (2*(1 - x)^22)/11 - (6*(1 - x)^23)/23 + (1 - x)^24/6 - (1 - x)^
25/25

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Rubi [A]  time = 0.0931179, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{1}{25} (1-x)^{25}+\frac{1}{6} (1-x)^{24}-\frac{6}{23} (1-x)^{23}+\frac{2}{11} (1-x)^{22}-\frac{1}{21} (1-x)^{21} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^20*x^4,x]

[Out]

-(1 - x)^21/21 + (2*(1 - x)^22)/11 - (6*(1 - x)^23)/23 + (1 - x)^24/6 - (1 - x)^
25/25

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Rubi in Sympy [A]  time = 7.44586, size = 36, normalized size = 0.64 \[ - \frac{\left (- x + 1\right )^{25}}{25} + \frac{\left (- x + 1\right )^{24}}{6} - \frac{6 \left (- x + 1\right )^{23}}{23} + \frac{2 \left (- x + 1\right )^{22}}{11} - \frac{\left (- x + 1\right )^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**20*x**4,x)

[Out]

-(-x + 1)**25/25 + (-x + 1)**24/6 - 6*(-x + 1)**23/23 + 2*(-x + 1)**22/11 - (-x
+ 1)**21/21

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Mathematica [B]  time = 0.00280465, size = 140, normalized size = 2.5 \[ \frac{x^{25}}{25}-\frac{5 x^{24}}{6}+\frac{190 x^{23}}{23}-\frac{570 x^{22}}{11}+\frac{1615 x^{21}}{7}-\frac{3876 x^{20}}{5}+2040 x^{19}-\frac{12920 x^{18}}{3}+7410 x^{17}-\frac{20995 x^{16}}{2}+\frac{184756 x^{15}}{15}-\frac{83980 x^{14}}{7}+9690 x^{13}-6460 x^{12}+\frac{38760 x^{11}}{11}-\frac{7752 x^{10}}{5}+\frac{1615 x^9}{3}-\frac{285 x^8}{2}+\frac{190 x^7}{7}-\frac{10 x^6}{3}+\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^20*x^4,x]

[Out]

x^5/5 - (10*x^6)/3 + (190*x^7)/7 - (285*x^8)/2 + (1615*x^9)/3 - (7752*x^10)/5 +
(38760*x^11)/11 - 6460*x^12 + 9690*x^13 - (83980*x^14)/7 + (184756*x^15)/15 - (2
0995*x^16)/2 + 7410*x^17 - (12920*x^18)/3 + 2040*x^19 - (3876*x^20)/5 + (1615*x^
21)/7 - (570*x^22)/11 + (190*x^23)/23 - (5*x^24)/6 + x^25/25

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Maple [B]  time = 0.003, size = 107, normalized size = 1.9 \[{\frac{{x}^{25}}{25}}-{\frac{5\,{x}^{24}}{6}}+{\frac{190\,{x}^{23}}{23}}-{\frac{570\,{x}^{22}}{11}}+{\frac{1615\,{x}^{21}}{7}}-{\frac{3876\,{x}^{20}}{5}}+2040\,{x}^{19}-{\frac{12920\,{x}^{18}}{3}}+7410\,{x}^{17}-{\frac{20995\,{x}^{16}}{2}}+{\frac{184756\,{x}^{15}}{15}}-{\frac{83980\,{x}^{14}}{7}}+9690\,{x}^{13}-6460\,{x}^{12}+{\frac{38760\,{x}^{11}}{11}}-{\frac{7752\,{x}^{10}}{5}}+{\frac{1615\,{x}^{9}}{3}}-{\frac{285\,{x}^{8}}{2}}+{\frac{190\,{x}^{7}}{7}}-{\frac{10\,{x}^{6}}{3}}+{\frac{{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^20*x^4,x)

[Out]

1/25*x^25-5/6*x^24+190/23*x^23-570/11*x^22+1615/7*x^21-3876/5*x^20+2040*x^19-129
20/3*x^18+7410*x^17-20995/2*x^16+184756/15*x^15-83980/7*x^14+9690*x^13-6460*x^12
+38760/11*x^11-7752/5*x^10+1615/3*x^9-285/2*x^8+190/7*x^7-10/3*x^6+1/5*x^5

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Maxima [A]  time = 1.37836, size = 143, normalized size = 2.55 \[ \frac{1}{25} \, x^{25} - \frac{5}{6} \, x^{24} + \frac{190}{23} \, x^{23} - \frac{570}{11} \, x^{22} + \frac{1615}{7} \, x^{21} - \frac{3876}{5} \, x^{20} + 2040 \, x^{19} - \frac{12920}{3} \, x^{18} + 7410 \, x^{17} - \frac{20995}{2} \, x^{16} + \frac{184756}{15} \, x^{15} - \frac{83980}{7} \, x^{14} + 9690 \, x^{13} - 6460 \, x^{12} + \frac{38760}{11} \, x^{11} - \frac{7752}{5} \, x^{10} + \frac{1615}{3} \, x^{9} - \frac{285}{2} \, x^{8} + \frac{190}{7} \, x^{7} - \frac{10}{3} \, x^{6} + \frac{1}{5} \, x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)^20*x^4,x, algorithm="maxima")

[Out]

1/25*x^25 - 5/6*x^24 + 190/23*x^23 - 570/11*x^22 + 1615/7*x^21 - 3876/5*x^20 + 2
040*x^19 - 12920/3*x^18 + 7410*x^17 - 20995/2*x^16 + 184756/15*x^15 - 83980/7*x^
14 + 9690*x^13 - 6460*x^12 + 38760/11*x^11 - 7752/5*x^10 + 1615/3*x^9 - 285/2*x^
8 + 190/7*x^7 - 10/3*x^6 + 1/5*x^5

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Fricas [A]  time = 0.178299, size = 1, normalized size = 0.02 \[ \frac{1}{25} x^{25} - \frac{5}{6} x^{24} + \frac{190}{23} x^{23} - \frac{570}{11} x^{22} + \frac{1615}{7} x^{21} - \frac{3876}{5} x^{20} + 2040 x^{19} - \frac{12920}{3} x^{18} + 7410 x^{17} - \frac{20995}{2} x^{16} + \frac{184756}{15} x^{15} - \frac{83980}{7} x^{14} + 9690 x^{13} - 6460 x^{12} + \frac{38760}{11} x^{11} - \frac{7752}{5} x^{10} + \frac{1615}{3} x^{9} - \frac{285}{2} x^{8} + \frac{190}{7} x^{7} - \frac{10}{3} x^{6} + \frac{1}{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)^20*x^4,x, algorithm="fricas")

[Out]

1/25*x^25 - 5/6*x^24 + 190/23*x^23 - 570/11*x^22 + 1615/7*x^21 - 3876/5*x^20 + 2
040*x^19 - 12920/3*x^18 + 7410*x^17 - 20995/2*x^16 + 184756/15*x^15 - 83980/7*x^
14 + 9690*x^13 - 6460*x^12 + 38760/11*x^11 - 7752/5*x^10 + 1615/3*x^9 - 285/2*x^
8 + 190/7*x^7 - 10/3*x^6 + 1/5*x^5

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Sympy [A]  time = 0.066715, size = 131, normalized size = 2.34 \[ \frac{x^{25}}{25} - \frac{5 x^{24}}{6} + \frac{190 x^{23}}{23} - \frac{570 x^{22}}{11} + \frac{1615 x^{21}}{7} - \frac{3876 x^{20}}{5} + 2040 x^{19} - \frac{12920 x^{18}}{3} + 7410 x^{17} - \frac{20995 x^{16}}{2} + \frac{184756 x^{15}}{15} - \frac{83980 x^{14}}{7} + 9690 x^{13} - 6460 x^{12} + \frac{38760 x^{11}}{11} - \frac{7752 x^{10}}{5} + \frac{1615 x^{9}}{3} - \frac{285 x^{8}}{2} + \frac{190 x^{7}}{7} - \frac{10 x^{6}}{3} + \frac{x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)**20*x**4,x)

[Out]

x**25/25 - 5*x**24/6 + 190*x**23/23 - 570*x**22/11 + 1615*x**21/7 - 3876*x**20/5
+ 2040*x**19 - 12920*x**18/3 + 7410*x**17 - 20995*x**16/2 + 184756*x**15/15 - 8
3980*x**14/7 + 9690*x**13 - 6460*x**12 + 38760*x**11/11 - 7752*x**10/5 + 1615*x*
*9/3 - 285*x**8/2 + 190*x**7/7 - 10*x**6/3 + x**5/5

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GIAC/XCAS [A]  time = 0.218893, size = 143, normalized size = 2.55 \[ \frac{1}{25} \, x^{25} - \frac{5}{6} \, x^{24} + \frac{190}{23} \, x^{23} - \frac{570}{11} \, x^{22} + \frac{1615}{7} \, x^{21} - \frac{3876}{5} \, x^{20} + 2040 \, x^{19} - \frac{12920}{3} \, x^{18} + 7410 \, x^{17} - \frac{20995}{2} \, x^{16} + \frac{184756}{15} \, x^{15} - \frac{83980}{7} \, x^{14} + 9690 \, x^{13} - 6460 \, x^{12} + \frac{38760}{11} \, x^{11} - \frac{7752}{5} \, x^{10} + \frac{1615}{3} \, x^{9} - \frac{285}{2} \, x^{8} + \frac{190}{7} \, x^{7} - \frac{10}{3} \, x^{6} + \frac{1}{5} \, x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)^20*x^4,x, algorithm="giac")

[Out]

1/25*x^25 - 5/6*x^24 + 190/23*x^23 - 570/11*x^22 + 1615/7*x^21 - 3876/5*x^20 + 2
040*x^19 - 12920/3*x^18 + 7410*x^17 - 20995/2*x^16 + 184756/15*x^15 - 83980/7*x^
14 + 9690*x^13 - 6460*x^12 + 38760/11*x^11 - 7752/5*x^10 + 1615/3*x^9 - 285/2*x^
8 + 190/7*x^7 - 10/3*x^6 + 1/5*x^5