3.129 \(\int x^5 (a+b x)^{10} \, dx\)

Optimal. Leaf size=98 \[ -\frac{a^5 (a+b x)^{11}}{11 b^6}+\frac{5 a^4 (a+b x)^{12}}{12 b^6}-\frac{10 a^3 (a+b x)^{13}}{13 b^6}+\frac{5 a^2 (a+b x)^{14}}{7 b^6}+\frac{(a+b x)^{16}}{16 b^6}-\frac{a (a+b x)^{15}}{3 b^6} \]

[Out]

-(a^5*(a + b*x)^11)/(11*b^6) + (5*a^4*(a + b*x)^12)/(12*b^6) - (10*a^3*(a + b*x)
^13)/(13*b^6) + (5*a^2*(a + b*x)^14)/(7*b^6) - (a*(a + b*x)^15)/(3*b^6) + (a + b
*x)^16/(16*b^6)

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Rubi [A]  time = 0.113513, antiderivative size = 98, 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 \[ -\frac{a^5 (a+b x)^{11}}{11 b^6}+\frac{5 a^4 (a+b x)^{12}}{12 b^6}-\frac{10 a^3 (a+b x)^{13}}{13 b^6}+\frac{5 a^2 (a+b x)^{14}}{7 b^6}+\frac{(a+b x)^{16}}{16 b^6}-\frac{a (a+b x)^{15}}{3 b^6} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x)^10,x]

[Out]

-(a^5*(a + b*x)^11)/(11*b^6) + (5*a^4*(a + b*x)^12)/(12*b^6) - (10*a^3*(a + b*x)
^13)/(13*b^6) + (5*a^2*(a + b*x)^14)/(7*b^6) - (a*(a + b*x)^15)/(3*b^6) + (a + b
*x)^16/(16*b^6)

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Rubi in Sympy [A]  time = 25.133, size = 90, normalized size = 0.92 \[ - \frac{a^{5} \left (a + b x\right )^{11}}{11 b^{6}} + \frac{5 a^{4} \left (a + b x\right )^{12}}{12 b^{6}} - \frac{10 a^{3} \left (a + b x\right )^{13}}{13 b^{6}} + \frac{5 a^{2} \left (a + b x\right )^{14}}{7 b^{6}} - \frac{a \left (a + b x\right )^{15}}{3 b^{6}} + \frac{\left (a + b x\right )^{16}}{16 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x+a)**10,x)

[Out]

-a**5*(a + b*x)**11/(11*b**6) + 5*a**4*(a + b*x)**12/(12*b**6) - 10*a**3*(a + b*
x)**13/(13*b**6) + 5*a**2*(a + b*x)**14/(7*b**6) - a*(a + b*x)**15/(3*b**6) + (a
+ b*x)**16/(16*b**6)

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Mathematica [A]  time = 0.00426473, size = 132, normalized size = 1.35 \[ \frac{a^{10} x^6}{6}+\frac{10}{7} a^9 b x^7+\frac{45}{8} a^8 b^2 x^8+\frac{40}{3} a^7 b^3 x^9+21 a^6 b^4 x^{10}+\frac{252}{11} a^5 b^5 x^{11}+\frac{35}{2} a^4 b^6 x^{12}+\frac{120}{13} a^3 b^7 x^{13}+\frac{45}{14} a^2 b^8 x^{14}+\frac{2}{3} a b^9 x^{15}+\frac{b^{10} x^{16}}{16} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x)^10,x]

[Out]

(a^10*x^6)/6 + (10*a^9*b*x^7)/7 + (45*a^8*b^2*x^8)/8 + (40*a^7*b^3*x^9)/3 + 21*a
^6*b^4*x^10 + (252*a^5*b^5*x^11)/11 + (35*a^4*b^6*x^12)/2 + (120*a^3*b^7*x^13)/1
3 + (45*a^2*b^8*x^14)/14 + (2*a*b^9*x^15)/3 + (b^10*x^16)/16

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Maple [A]  time = 0.003, size = 113, normalized size = 1.2 \[{\frac{{b}^{10}{x}^{16}}{16}}+{\frac{2\,a{b}^{9}{x}^{15}}{3}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{14}}{14}}+{\frac{120\,{a}^{3}{b}^{7}{x}^{13}}{13}}+{\frac{35\,{a}^{4}{b}^{6}{x}^{12}}{2}}+{\frac{252\,{a}^{5}{b}^{5}{x}^{11}}{11}}+21\,{a}^{6}{b}^{4}{x}^{10}+{\frac{40\,{a}^{7}{b}^{3}{x}^{9}}{3}}+{\frac{45\,{a}^{8}{b}^{2}{x}^{8}}{8}}+{\frac{10\,{a}^{9}b{x}^{7}}{7}}+{\frac{{a}^{10}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x+a)^10,x)

[Out]

1/16*b^10*x^16+2/3*a*b^9*x^15+45/14*a^2*b^8*x^14+120/13*a^3*b^7*x^13+35/2*a^4*b^
6*x^12+252/11*a^5*b^5*x^11+21*a^6*b^4*x^10+40/3*a^7*b^3*x^9+45/8*a^8*b^2*x^8+10/
7*a^9*b*x^7+1/6*a^10*x^6

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Maxima [A]  time = 1.36262, size = 151, normalized size = 1.54 \[ \frac{1}{16} \, b^{10} x^{16} + \frac{2}{3} \, a b^{9} x^{15} + \frac{45}{14} \, a^{2} b^{8} x^{14} + \frac{120}{13} \, a^{3} b^{7} x^{13} + \frac{35}{2} \, a^{4} b^{6} x^{12} + \frac{252}{11} \, a^{5} b^{5} x^{11} + 21 \, a^{6} b^{4} x^{10} + \frac{40}{3} \, a^{7} b^{3} x^{9} + \frac{45}{8} \, a^{8} b^{2} x^{8} + \frac{10}{7} \, a^{9} b x^{7} + \frac{1}{6} \, a^{10} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^5,x, algorithm="maxima")

[Out]

1/16*b^10*x^16 + 2/3*a*b^9*x^15 + 45/14*a^2*b^8*x^14 + 120/13*a^3*b^7*x^13 + 35/
2*a^4*b^6*x^12 + 252/11*a^5*b^5*x^11 + 21*a^6*b^4*x^10 + 40/3*a^7*b^3*x^9 + 45/8
*a^8*b^2*x^8 + 10/7*a^9*b*x^7 + 1/6*a^10*x^6

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Fricas [A]  time = 0.173261, size = 1, normalized size = 0.01 \[ \frac{1}{16} x^{16} b^{10} + \frac{2}{3} x^{15} b^{9} a + \frac{45}{14} x^{14} b^{8} a^{2} + \frac{120}{13} x^{13} b^{7} a^{3} + \frac{35}{2} x^{12} b^{6} a^{4} + \frac{252}{11} x^{11} b^{5} a^{5} + 21 x^{10} b^{4} a^{6} + \frac{40}{3} x^{9} b^{3} a^{7} + \frac{45}{8} x^{8} b^{2} a^{8} + \frac{10}{7} x^{7} b a^{9} + \frac{1}{6} x^{6} a^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^5,x, algorithm="fricas")

[Out]

1/16*x^16*b^10 + 2/3*x^15*b^9*a + 45/14*x^14*b^8*a^2 + 120/13*x^13*b^7*a^3 + 35/
2*x^12*b^6*a^4 + 252/11*x^11*b^5*a^5 + 21*x^10*b^4*a^6 + 40/3*x^9*b^3*a^7 + 45/8
*x^8*b^2*a^8 + 10/7*x^7*b*a^9 + 1/6*x^6*a^10

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Sympy [A]  time = 0.177077, size = 133, normalized size = 1.36 \[ \frac{a^{10} x^{6}}{6} + \frac{10 a^{9} b x^{7}}{7} + \frac{45 a^{8} b^{2} x^{8}}{8} + \frac{40 a^{7} b^{3} x^{9}}{3} + 21 a^{6} b^{4} x^{10} + \frac{252 a^{5} b^{5} x^{11}}{11} + \frac{35 a^{4} b^{6} x^{12}}{2} + \frac{120 a^{3} b^{7} x^{13}}{13} + \frac{45 a^{2} b^{8} x^{14}}{14} + \frac{2 a b^{9} x^{15}}{3} + \frac{b^{10} x^{16}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x+a)**10,x)

[Out]

a**10*x**6/6 + 10*a**9*b*x**7/7 + 45*a**8*b**2*x**8/8 + 40*a**7*b**3*x**9/3 + 21
*a**6*b**4*x**10 + 252*a**5*b**5*x**11/11 + 35*a**4*b**6*x**12/2 + 120*a**3*b**7
*x**13/13 + 45*a**2*b**8*x**14/14 + 2*a*b**9*x**15/3 + b**10*x**16/16

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GIAC/XCAS [A]  time = 0.203946, size = 151, normalized size = 1.54 \[ \frac{1}{16} \, b^{10} x^{16} + \frac{2}{3} \, a b^{9} x^{15} + \frac{45}{14} \, a^{2} b^{8} x^{14} + \frac{120}{13} \, a^{3} b^{7} x^{13} + \frac{35}{2} \, a^{4} b^{6} x^{12} + \frac{252}{11} \, a^{5} b^{5} x^{11} + 21 \, a^{6} b^{4} x^{10} + \frac{40}{3} \, a^{7} b^{3} x^{9} + \frac{45}{8} \, a^{8} b^{2} x^{8} + \frac{10}{7} \, a^{9} b x^{7} + \frac{1}{6} \, a^{10} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^5,x, algorithm="giac")

[Out]

1/16*b^10*x^16 + 2/3*a*b^9*x^15 + 45/14*a^2*b^8*x^14 + 120/13*a^3*b^7*x^13 + 35/
2*a^4*b^6*x^12 + 252/11*a^5*b^5*x^11 + 21*a^6*b^4*x^10 + 40/3*a^7*b^3*x^9 + 45/8
*a^8*b^2*x^8 + 10/7*a^9*b*x^7 + 1/6*a^10*x^6