3.1.51 \(\int (1-x)^{20} x^4 \, dx\) [51]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^20*x^4,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (1-x)^{20} x^4 \, dx &=\int \left ((1-x)^{20}-4 (1-x)^{21}+6 (1-x)^{22}-4 (1-x)^{23}+(1-x)^{24}\right ) \, dx\\ &=-\frac {1}{21} (1-x)^{21}+\frac {2}{11} (1-x)^{22}-\frac {6}{23} (1-x)^{23}+\frac {1}{6} (1-x)^{24}-\frac {1}{25} (1-x)^{25}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(56)=112\).
time = 0.00, size = 140, normalized size = 2.50 \begin {gather*} \frac {x^5}{5}-\frac {10 x^6}{3}+\frac {190 x^7}{7}-\frac {285 x^8}{2}+\frac {1615 x^9}{3}-\frac {7752 x^{10}}{5}+\frac {38760 x^{11}}{11}-6460 x^{12}+9690 x^{13}-\frac {83980 x^{14}}{7}+\frac {184756 x^{15}}{15}-\frac {20995 x^{16}}{2}+7410 x^{17}-\frac {12920 x^{18}}{3}+2040 x^{19}-\frac {3876 x^{20}}{5}+\frac {1615 x^{21}}{7}-\frac {570 x^{22}}{11}+\frac {190 x^{23}}{23}-\frac {5 x^{24}}{6}+\frac {x^{25}}{25} \end {gather*}

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 - (20995*x^16)/2 + 7410*x^17 - (12920*x^18)/3 + 2040*x^19 - (387
6*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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Mathics [A]
time = 2.05, size = 105, normalized size = 1.88 \begin {gather*} \frac {x^5 \left (53130-885500 x+7210500 x^2-37855125 x^3+143008250 x^4-411863760 x^5+936054000 x^6-1716099000 x^7+2574148500 x^8-3187041000 x^9+3272028760 x^{10}-2788660875 x^{11}[phone] x^{12}-1144066000 x^{13}[phone] x^{14}-205931880 x^{15}[phone] x^{16}-13765500 x^{17}+2194500 x^{18}-221375 x^{19}+10626 x^{20}\right )}{265650} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^4*(1 - x)^20,x]')

[Out]

x ^ 5 (53130 - 885500 x + 7210500 x ^ 2 - 37855125 x ^ 3 + 143008250 x ^ 4 - 411863760 x ^ 5 + 936054000 x ^ 6
- 1716099000 x ^ 7 + 2574148500 x ^ 8 - 3187041000 x ^ 9 + 3272028760 x ^ 10 - 2788660875 x ^ 11 + 1968466500
x ^ 12 - 1144066000 x ^ 13 + 541926000 x ^ 14 - 205931880 x ^ 15 + 61289250 x ^ 16 - 13765500 x ^ 17 + 219450
0 x ^ 18 - 221375 x ^ 19 + 10626 x ^ 20) / 265650

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs. \(2(46)=92\).
time = 0.07, size = 107, normalized size = 1.91

method	result	size
gosper	\(\frac {x^{5} \left (10626 x^{20}-221375 x^{19}+2194500 x^{18}-13765500 x^{17}[phone] x^{16}-205931880 x^{15}[phone] x^{14}-1144066000 x^{13}[phone] x^{12}-2788660875 x^{11}[phone] x^{10}-3187041000 x^{9}[phone] x^{8}-1716099000 x^{7}[phone] x^{6}-411863760 x^{5}[phone] x^{4}-37855125 x^{3}+7210500 x^{2}-885500 x +53130\right )}{265650}\)	\(106\)
default	\(-6460 x^{12}+9690 x^{13}-\frac {83980}{7} x^{14}+\frac {184756}{15} x^{15}-\frac {20995}{2} x^{16}+7410 x^{17}-\frac {12920}{3} x^{18}+2040 x^{19}-\frac {3876}{5} x^{20}+\frac {1615}{7} x^{21}-\frac {570}{11} x^{22}+\frac {190}{23} x^{23}-\frac {5}{6} x^{24}+\frac {1}{25} x^{25}+\frac {1615}{3} x^{9}-\frac {7752}{5} x^{10}+\frac {38760}{11} x^{11}-\frac {10}{3} x^{6}+\frac {190}{7} x^{7}-\frac {285}{2} x^{8}+\frac {1}{5} x^{5}\)	\(107\)
norman	\(-6460 x^{12}+9690 x^{13}-\frac {83980}{7} x^{14}+\frac {184756}{15} x^{15}-\frac {20995}{2} x^{16}+7410 x^{17}-\frac {12920}{3} x^{18}+2040 x^{19}-\frac {3876}{5} x^{20}+\frac {1615}{7} x^{21}-\frac {570}{11} x^{22}+\frac {190}{23} x^{23}-\frac {5}{6} x^{24}+\frac {1}{25} x^{25}+\frac {1615}{3} x^{9}-\frac {7752}{5} x^{10}+\frac {38760}{11} x^{11}-\frac {10}{3} x^{6}+\frac {190}{7} x^{7}-\frac {285}{2} x^{8}+\frac {1}{5} x^{5}\)	\(107\)
risch	\(-6460 x^{12}+9690 x^{13}-\frac {83980}{7} x^{14}+\frac {184756}{15} x^{15}-\frac {20995}{2} x^{16}+7410 x^{17}-\frac {12920}{3} x^{18}+2040 x^{19}-\frac {3876}{5} x^{20}+\frac {1615}{7} x^{21}-\frac {570}{11} x^{22}+\frac {190}{23} x^{23}-\frac {5}{6} x^{24}+\frac {1}{25} x^{25}+\frac {1615}{3} x^{9}-\frac {7752}{5} x^{10}+\frac {38760}{11} x^{11}-\frac {10}{3} x^{6}+\frac {190}{7} x^{7}-\frac {285}{2} x^{8}+\frac {1}{5} x^{5}\)	\(107\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^20*x^4,x,method=_RETURNVERBOSE)

[Out]

-6460*x^12+9690*x^13-83980/7*x^14+184756/15*x^15-20995/2*x^16+7410*x^17-12920/3*x^18+2040*x^19-3876/5*x^20+161
5/7*x^21-570/11*x^22+190/23*x^23-5/6*x^24+1/25*x^25+1615/3*x^9-7752/5*x^10+38760/11*x^11-10/3*x^6+190/7*x^7-28
5/2*x^8+1/5*x^5

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (36) = 72\).
time = 0.29, size = 106, normalized size = 1.89 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^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 + 2040*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 + 1
615/3*x^9 - 285/2*x^8 + 190/7*x^7 - 10/3*x^6 + 1/5*x^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (36) = 72\).
time = 0.37, size = 106, normalized size = 1.89 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^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 + 2040*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 + 1
615/3*x^9 - 285/2*x^8 + 190/7*x^7 - 10/3*x^6 + 1/5*x^5

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (36) = 72\)
time = 0.04, size = 131, normalized size = 2.34 \begin {gather*} \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} \end {gather*}

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 - 83980*x**14/7 + 9690*x**13 - 6460*x**12 + 38760*x**11/11 - 77
52*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 [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (36) = 72\).
time = 0.00, size = 150, normalized size = 2.68 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 - 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 + 1
615/3*x^9 - 285/2*x^8 + 190/7*x^7 - 10/3*x^6 + 1/5*x^5

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Mupad [B]
time = 0.46, size = 106, normalized size = 1.89 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(x - 1)^20,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 - (20995*x^16)/2 + 7410*x^17 - (12920*x^18)/3 + 2040*x^19 - (387
6*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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