3.123 \(\int x^{11} (a+b x)^{10} \, dx\)

Optimal. Leaf size=132 \[ \frac{a^{10} x^{12}}{12}+\frac{10}{13} a^9 b x^{13}+\frac{45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac{105}{8} a^6 b^4 x^{16}+\frac{252}{17} a^5 b^5 x^{17}+\frac{35}{3} a^4 b^6 x^{18}+\frac{120}{19} a^3 b^7 x^{19}+\frac{9}{4} a^2 b^8 x^{20}+\frac{10}{21} a b^9 x^{21}+\frac{b^{10} x^{22}}{22} \]

[Out]

(a^10*x^12)/12 + (10*a^9*b*x^13)/13 + (45*a^8*b^2*x^14)/14 + 8*a^7*b^3*x^15 + (1
05*a^6*b^4*x^16)/8 + (252*a^5*b^5*x^17)/17 + (35*a^4*b^6*x^18)/3 + (120*a^3*b^7*
x^19)/19 + (9*a^2*b^8*x^20)/4 + (10*a*b^9*x^21)/21 + (b^10*x^22)/22

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Rubi [A]  time = 0.161009, antiderivative size = 132, 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 \[ \frac{a^{10} x^{12}}{12}+\frac{10}{13} a^9 b x^{13}+\frac{45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac{105}{8} a^6 b^4 x^{16}+\frac{252}{17} a^5 b^5 x^{17}+\frac{35}{3} a^4 b^6 x^{18}+\frac{120}{19} a^3 b^7 x^{19}+\frac{9}{4} a^2 b^8 x^{20}+\frac{10}{21} a b^9 x^{21}+\frac{b^{10} x^{22}}{22} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a + b*x)^10,x]

[Out]

(a^10*x^12)/12 + (10*a^9*b*x^13)/13 + (45*a^8*b^2*x^14)/14 + 8*a^7*b^3*x^15 + (1
05*a^6*b^4*x^16)/8 + (252*a^5*b^5*x^17)/17 + (35*a^4*b^6*x^18)/3 + (120*a^3*b^7*
x^19)/19 + (9*a^2*b^8*x^20)/4 + (10*a*b^9*x^21)/21 + (b^10*x^22)/22

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Rubi in Sympy [A]  time = 28.9923, size = 133, normalized size = 1.01 \[ \frac{a^{10} x^{12}}{12} + \frac{10 a^{9} b x^{13}}{13} + \frac{45 a^{8} b^{2} x^{14}}{14} + 8 a^{7} b^{3} x^{15} + \frac{105 a^{6} b^{4} x^{16}}{8} + \frac{252 a^{5} b^{5} x^{17}}{17} + \frac{35 a^{4} b^{6} x^{18}}{3} + \frac{120 a^{3} b^{7} x^{19}}{19} + \frac{9 a^{2} b^{8} x^{20}}{4} + \frac{10 a b^{9} x^{21}}{21} + \frac{b^{10} x^{22}}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(b*x+a)**10,x)

[Out]

a**10*x**12/12 + 10*a**9*b*x**13/13 + 45*a**8*b**2*x**14/14 + 8*a**7*b**3*x**15
+ 105*a**6*b**4*x**16/8 + 252*a**5*b**5*x**17/17 + 35*a**4*b**6*x**18/3 + 120*a*
*3*b**7*x**19/19 + 9*a**2*b**8*x**20/4 + 10*a*b**9*x**21/21 + b**10*x**22/22

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Mathematica [A]  time = 0.0045908, size = 132, normalized size = 1. \[ \frac{a^{10} x^{12}}{12}+\frac{10}{13} a^9 b x^{13}+\frac{45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac{105}{8} a^6 b^4 x^{16}+\frac{252}{17} a^5 b^5 x^{17}+\frac{35}{3} a^4 b^6 x^{18}+\frac{120}{19} a^3 b^7 x^{19}+\frac{9}{4} a^2 b^8 x^{20}+\frac{10}{21} a b^9 x^{21}+\frac{b^{10} x^{22}}{22} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a + b*x)^10,x]

[Out]

(a^10*x^12)/12 + (10*a^9*b*x^13)/13 + (45*a^8*b^2*x^14)/14 + 8*a^7*b^3*x^15 + (1
05*a^6*b^4*x^16)/8 + (252*a^5*b^5*x^17)/17 + (35*a^4*b^6*x^18)/3 + (120*a^3*b^7*
x^19)/19 + (9*a^2*b^8*x^20)/4 + (10*a*b^9*x^21)/21 + (b^10*x^22)/22

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Maple [A]  time = 0.003, size = 113, normalized size = 0.9 \[{\frac{{a}^{10}{x}^{12}}{12}}+{\frac{10\,{a}^{9}b{x}^{13}}{13}}+{\frac{45\,{a}^{8}{b}^{2}{x}^{14}}{14}}+8\,{a}^{7}{b}^{3}{x}^{15}+{\frac{105\,{a}^{6}{b}^{4}{x}^{16}}{8}}+{\frac{252\,{a}^{5}{b}^{5}{x}^{17}}{17}}+{\frac{35\,{a}^{4}{b}^{6}{x}^{18}}{3}}+{\frac{120\,{a}^{3}{b}^{7}{x}^{19}}{19}}+{\frac{9\,{a}^{2}{b}^{8}{x}^{20}}{4}}+{\frac{10\,a{b}^{9}{x}^{21}}{21}}+{\frac{{b}^{10}{x}^{22}}{22}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(b*x+a)^10,x)

[Out]

1/12*a^10*x^12+10/13*a^9*b*x^13+45/14*a^8*b^2*x^14+8*a^7*b^3*x^15+105/8*a^6*b^4*
x^16+252/17*a^5*b^5*x^17+35/3*a^4*b^6*x^18+120/19*a^3*b^7*x^19+9/4*a^2*b^8*x^20+
10/21*a*b^9*x^21+1/22*b^10*x^22

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Maxima [A]  time = 1.36386, size = 151, normalized size = 1.14 \[ \frac{1}{22} \, b^{10} x^{22} + \frac{10}{21} \, a b^{9} x^{21} + \frac{9}{4} \, a^{2} b^{8} x^{20} + \frac{120}{19} \, a^{3} b^{7} x^{19} + \frac{35}{3} \, a^{4} b^{6} x^{18} + \frac{252}{17} \, a^{5} b^{5} x^{17} + \frac{105}{8} \, a^{6} b^{4} x^{16} + 8 \, a^{7} b^{3} x^{15} + \frac{45}{14} \, a^{8} b^{2} x^{14} + \frac{10}{13} \, a^{9} b x^{13} + \frac{1}{12} \, a^{10} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^11,x, algorithm="maxima")

[Out]

1/22*b^10*x^22 + 10/21*a*b^9*x^21 + 9/4*a^2*b^8*x^20 + 120/19*a^3*b^7*x^19 + 35/
3*a^4*b^6*x^18 + 252/17*a^5*b^5*x^17 + 105/8*a^6*b^4*x^16 + 8*a^7*b^3*x^15 + 45/
14*a^8*b^2*x^14 + 10/13*a^9*b*x^13 + 1/12*a^10*x^12

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Fricas [A]  time = 0.174301, size = 1, normalized size = 0.01 \[ \frac{1}{22} x^{22} b^{10} + \frac{10}{21} x^{21} b^{9} a + \frac{9}{4} x^{20} b^{8} a^{2} + \frac{120}{19} x^{19} b^{7} a^{3} + \frac{35}{3} x^{18} b^{6} a^{4} + \frac{252}{17} x^{17} b^{5} a^{5} + \frac{105}{8} x^{16} b^{4} a^{6} + 8 x^{15} b^{3} a^{7} + \frac{45}{14} x^{14} b^{2} a^{8} + \frac{10}{13} x^{13} b a^{9} + \frac{1}{12} x^{12} a^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^11,x, algorithm="fricas")

[Out]

1/22*x^22*b^10 + 10/21*x^21*b^9*a + 9/4*x^20*b^8*a^2 + 120/19*x^19*b^7*a^3 + 35/
3*x^18*b^6*a^4 + 252/17*x^17*b^5*a^5 + 105/8*x^16*b^4*a^6 + 8*x^15*b^3*a^7 + 45/
14*x^14*b^2*a^8 + 10/13*x^13*b*a^9 + 1/12*x^12*a^10

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Sympy [A]  time = 0.174201, size = 133, normalized size = 1.01 \[ \frac{a^{10} x^{12}}{12} + \frac{10 a^{9} b x^{13}}{13} + \frac{45 a^{8} b^{2} x^{14}}{14} + 8 a^{7} b^{3} x^{15} + \frac{105 a^{6} b^{4} x^{16}}{8} + \frac{252 a^{5} b^{5} x^{17}}{17} + \frac{35 a^{4} b^{6} x^{18}}{3} + \frac{120 a^{3} b^{7} x^{19}}{19} + \frac{9 a^{2} b^{8} x^{20}}{4} + \frac{10 a b^{9} x^{21}}{21} + \frac{b^{10} x^{22}}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(b*x+a)**10,x)

[Out]

a**10*x**12/12 + 10*a**9*b*x**13/13 + 45*a**8*b**2*x**14/14 + 8*a**7*b**3*x**15
+ 105*a**6*b**4*x**16/8 + 252*a**5*b**5*x**17/17 + 35*a**4*b**6*x**18/3 + 120*a*
*3*b**7*x**19/19 + 9*a**2*b**8*x**20/4 + 10*a*b**9*x**21/21 + b**10*x**22/22

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GIAC/XCAS [A]  time = 0.211898, size = 151, normalized size = 1.14 \[ \frac{1}{22} \, b^{10} x^{22} + \frac{10}{21} \, a b^{9} x^{21} + \frac{9}{4} \, a^{2} b^{8} x^{20} + \frac{120}{19} \, a^{3} b^{7} x^{19} + \frac{35}{3} \, a^{4} b^{6} x^{18} + \frac{252}{17} \, a^{5} b^{5} x^{17} + \frac{105}{8} \, a^{6} b^{4} x^{16} + 8 \, a^{7} b^{3} x^{15} + \frac{45}{14} \, a^{8} b^{2} x^{14} + \frac{10}{13} \, a^{9} b x^{13} + \frac{1}{12} \, a^{10} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^11,x, algorithm="giac")

[Out]

1/22*b^10*x^22 + 10/21*a*b^9*x^21 + 9/4*a^2*b^8*x^20 + 120/19*a^3*b^7*x^19 + 35/
3*a^4*b^6*x^18 + 252/17*a^5*b^5*x^17 + 105/8*a^6*b^4*x^16 + 8*a^7*b^3*x^15 + 45/
14*a^8*b^2*x^14 + 10/13*a^9*b*x^13 + 1/12*a^10*x^12