3.125 \(\int x^9 (a+b x)^{10} \, dx\)

Optimal. Leaf size=132 \[ \frac{a^{10} x^{10}}{10}+\frac{10}{11} a^9 b x^{11}+\frac{15}{4} a^8 b^2 x^{12}+\frac{120}{13} a^7 b^3 x^{13}+15 a^6 b^4 x^{14}+\frac{84}{5} a^5 b^5 x^{15}+\frac{105}{8} a^4 b^6 x^{16}+\frac{120}{17} a^3 b^7 x^{17}+\frac{5}{2} a^2 b^8 x^{18}+\frac{10}{19} a b^9 x^{19}+\frac{b^{10} x^{20}}{20} \]

[Out]

(a^10*x^10)/10 + (10*a^9*b*x^11)/11 + (15*a^8*b^2*x^12)/4 + (120*a^7*b^3*x^13)/1
3 + 15*a^6*b^4*x^14 + (84*a^5*b^5*x^15)/5 + (105*a^4*b^6*x^16)/8 + (120*a^3*b^7*
x^17)/17 + (5*a^2*b^8*x^18)/2 + (10*a*b^9*x^19)/19 + (b^10*x^20)/20

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Rubi [A]  time = 0.146712, 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 \[ \frac{a^{10} x^{10}}{10}+\frac{10}{11} a^9 b x^{11}+\frac{15}{4} a^8 b^2 x^{12}+\frac{120}{13} a^7 b^3 x^{13}+15 a^6 b^4 x^{14}+\frac{84}{5} a^5 b^5 x^{15}+\frac{105}{8} a^4 b^6 x^{16}+\frac{120}{17} a^3 b^7 x^{17}+\frac{5}{2} a^2 b^8 x^{18}+\frac{10}{19} a b^9 x^{19}+\frac{b^{10} x^{20}}{20} \]

Antiderivative was successfully verified.

[In]  Int[x^9*(a + b*x)^10,x]

[Out]

(a^10*x^10)/10 + (10*a^9*b*x^11)/11 + (15*a^8*b^2*x^12)/4 + (120*a^7*b^3*x^13)/1
3 + 15*a^6*b^4*x^14 + (84*a^5*b^5*x^15)/5 + (105*a^4*b^6*x^16)/8 + (120*a^3*b^7*
x^17)/17 + (5*a^2*b^8*x^18)/2 + (10*a*b^9*x^19)/19 + (b^10*x^20)/20

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Rubi in Sympy [A]  time = 27.5103, size = 133, normalized size = 1.01 \[ \frac{a^{10} x^{10}}{10} + \frac{10 a^{9} b x^{11}}{11} + \frac{15 a^{8} b^{2} x^{12}}{4} + \frac{120 a^{7} b^{3} x^{13}}{13} + 15 a^{6} b^{4} x^{14} + \frac{84 a^{5} b^{5} x^{15}}{5} + \frac{105 a^{4} b^{6} x^{16}}{8} + \frac{120 a^{3} b^{7} x^{17}}{17} + \frac{5 a^{2} b^{8} x^{18}}{2} + \frac{10 a b^{9} x^{19}}{19} + \frac{b^{10} x^{20}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(b*x+a)**10,x)

[Out]

a**10*x**10/10 + 10*a**9*b*x**11/11 + 15*a**8*b**2*x**12/4 + 120*a**7*b**3*x**13
/13 + 15*a**6*b**4*x**14 + 84*a**5*b**5*x**15/5 + 105*a**4*b**6*x**16/8 + 120*a*
*3*b**7*x**17/17 + 5*a**2*b**8*x**18/2 + 10*a*b**9*x**19/19 + b**10*x**20/20

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Mathematica [A]  time = 0.00426473, size = 132, normalized size = 1. \[ \frac{a^{10} x^{10}}{10}+\frac{10}{11} a^9 b x^{11}+\frac{15}{4} a^8 b^2 x^{12}+\frac{120}{13} a^7 b^3 x^{13}+15 a^6 b^4 x^{14}+\frac{84}{5} a^5 b^5 x^{15}+\frac{105}{8} a^4 b^6 x^{16}+\frac{120}{17} a^3 b^7 x^{17}+\frac{5}{2} a^2 b^8 x^{18}+\frac{10}{19} a b^9 x^{19}+\frac{b^{10} x^{20}}{20} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9*(a + b*x)^10,x]

[Out]

(a^10*x^10)/10 + (10*a^9*b*x^11)/11 + (15*a^8*b^2*x^12)/4 + (120*a^7*b^3*x^13)/1
3 + 15*a^6*b^4*x^14 + (84*a^5*b^5*x^15)/5 + (105*a^4*b^6*x^16)/8 + (120*a^3*b^7*
x^17)/17 + (5*a^2*b^8*x^18)/2 + (10*a*b^9*x^19)/19 + (b^10*x^20)/20

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Maple [A]  time = 0.002, size = 113, normalized size = 0.9 \[{\frac{{a}^{10}{x}^{10}}{10}}+{\frac{10\,{a}^{9}b{x}^{11}}{11}}+{\frac{15\,{a}^{8}{b}^{2}{x}^{12}}{4}}+{\frac{120\,{a}^{7}{b}^{3}{x}^{13}}{13}}+15\,{a}^{6}{b}^{4}{x}^{14}+{\frac{84\,{a}^{5}{b}^{5}{x}^{15}}{5}}+{\frac{105\,{a}^{4}{b}^{6}{x}^{16}}{8}}+{\frac{120\,{a}^{3}{b}^{7}{x}^{17}}{17}}+{\frac{5\,{a}^{2}{b}^{8}{x}^{18}}{2}}+{\frac{10\,a{b}^{9}{x}^{19}}{19}}+{\frac{{b}^{10}{x}^{20}}{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(b*x+a)^10,x)

[Out]

1/10*a^10*x^10+10/11*a^9*b*x^11+15/4*a^8*b^2*x^12+120/13*a^7*b^3*x^13+15*a^6*b^4
*x^14+84/5*a^5*b^5*x^15+105/8*a^4*b^6*x^16+120/17*a^3*b^7*x^17+5/2*a^2*b^8*x^18+
10/19*a*b^9*x^19+1/20*b^10*x^20

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Maxima [A]  time = 1.34639, size = 151, normalized size = 1.14 \[ \frac{1}{20} \, b^{10} x^{20} + \frac{10}{19} \, a b^{9} x^{19} + \frac{5}{2} \, a^{2} b^{8} x^{18} + \frac{120}{17} \, a^{3} b^{7} x^{17} + \frac{105}{8} \, a^{4} b^{6} x^{16} + \frac{84}{5} \, a^{5} b^{5} x^{15} + 15 \, a^{6} b^{4} x^{14} + \frac{120}{13} \, a^{7} b^{3} x^{13} + \frac{15}{4} \, a^{8} b^{2} x^{12} + \frac{10}{11} \, a^{9} b x^{11} + \frac{1}{10} \, a^{10} x^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^9,x, algorithm="maxima")

[Out]

1/20*b^10*x^20 + 10/19*a*b^9*x^19 + 5/2*a^2*b^8*x^18 + 120/17*a^3*b^7*x^17 + 105
/8*a^4*b^6*x^16 + 84/5*a^5*b^5*x^15 + 15*a^6*b^4*x^14 + 120/13*a^7*b^3*x^13 + 15
/4*a^8*b^2*x^12 + 10/11*a^9*b*x^11 + 1/10*a^10*x^10

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Fricas [A]  time = 0.179298, size = 1, normalized size = 0.01 \[ \frac{1}{20} x^{20} b^{10} + \frac{10}{19} x^{19} b^{9} a + \frac{5}{2} x^{18} b^{8} a^{2} + \frac{120}{17} x^{17} b^{7} a^{3} + \frac{105}{8} x^{16} b^{6} a^{4} + \frac{84}{5} x^{15} b^{5} a^{5} + 15 x^{14} b^{4} a^{6} + \frac{120}{13} x^{13} b^{3} a^{7} + \frac{15}{4} x^{12} b^{2} a^{8} + \frac{10}{11} x^{11} b a^{9} + \frac{1}{10} x^{10} a^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^9,x, algorithm="fricas")

[Out]

1/20*x^20*b^10 + 10/19*x^19*b^9*a + 5/2*x^18*b^8*a^2 + 120/17*x^17*b^7*a^3 + 105
/8*x^16*b^6*a^4 + 84/5*x^15*b^5*a^5 + 15*x^14*b^4*a^6 + 120/13*x^13*b^3*a^7 + 15
/4*x^12*b^2*a^8 + 10/11*x^11*b*a^9 + 1/10*x^10*a^10

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Sympy [A]  time = 0.176864, size = 133, normalized size = 1.01 \[ \frac{a^{10} x^{10}}{10} + \frac{10 a^{9} b x^{11}}{11} + \frac{15 a^{8} b^{2} x^{12}}{4} + \frac{120 a^{7} b^{3} x^{13}}{13} + 15 a^{6} b^{4} x^{14} + \frac{84 a^{5} b^{5} x^{15}}{5} + \frac{105 a^{4} b^{6} x^{16}}{8} + \frac{120 a^{3} b^{7} x^{17}}{17} + \frac{5 a^{2} b^{8} x^{18}}{2} + \frac{10 a b^{9} x^{19}}{19} + \frac{b^{10} x^{20}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(b*x+a)**10,x)

[Out]

a**10*x**10/10 + 10*a**9*b*x**11/11 + 15*a**8*b**2*x**12/4 + 120*a**7*b**3*x**13
/13 + 15*a**6*b**4*x**14 + 84*a**5*b**5*x**15/5 + 105*a**4*b**6*x**16/8 + 120*a*
*3*b**7*x**17/17 + 5*a**2*b**8*x**18/2 + 10*a*b**9*x**19/19 + b**10*x**20/20

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GIAC/XCAS [A]  time = 0.203144, size = 151, normalized size = 1.14 \[ \frac{1}{20} \, b^{10} x^{20} + \frac{10}{19} \, a b^{9} x^{19} + \frac{5}{2} \, a^{2} b^{8} x^{18} + \frac{120}{17} \, a^{3} b^{7} x^{17} + \frac{105}{8} \, a^{4} b^{6} x^{16} + \frac{84}{5} \, a^{5} b^{5} x^{15} + 15 \, a^{6} b^{4} x^{14} + \frac{120}{13} \, a^{7} b^{3} x^{13} + \frac{15}{4} \, a^{8} b^{2} x^{12} + \frac{10}{11} \, a^{9} b x^{11} + \frac{1}{10} \, a^{10} x^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^9,x, algorithm="giac")

[Out]

1/20*b^10*x^20 + 10/19*a*b^9*x^19 + 5/2*a^2*b^8*x^18 + 120/17*a^3*b^7*x^17 + 105
/8*a^4*b^6*x^16 + 84/5*a^5*b^5*x^15 + 15*a^6*b^4*x^14 + 120/13*a^7*b^3*x^13 + 15
/4*a^8*b^2*x^12 + 10/11*a^9*b*x^11 + 1/10*a^10*x^10