3.101 \(\int x^5 (a+b x)^7 \, dx\)

Optimal. Leaf size=96 \[ -\frac{a^5 (a+b x)^8}{8 b^6}+\frac{5 a^4 (a+b x)^9}{9 b^6}-\frac{a^3 (a+b x)^{10}}{b^6}+\frac{10 a^2 (a+b x)^{11}}{11 b^6}+\frac{(a+b x)^{13}}{13 b^6}-\frac{5 a (a+b x)^{12}}{12 b^6} \]

[Out]

-(a^5*(a + b*x)^8)/(8*b^6) + (5*a^4*(a + b*x)^9)/(9*b^6) - (a^3*(a + b*x)^10)/b^
6 + (10*a^2*(a + b*x)^11)/(11*b^6) - (5*a*(a + b*x)^12)/(12*b^6) + (a + b*x)^13/
(13*b^6)

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Rubi [A]  time = 0.0975971, antiderivative size = 96, 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 \[ -\frac{a^5 (a+b x)^8}{8 b^6}+\frac{5 a^4 (a+b x)^9}{9 b^6}-\frac{a^3 (a+b x)^{10}}{b^6}+\frac{10 a^2 (a+b x)^{11}}{11 b^6}+\frac{(a+b x)^{13}}{13 b^6}-\frac{5 a (a+b x)^{12}}{12 b^6} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x)^7,x]

[Out]

-(a^5*(a + b*x)^8)/(8*b^6) + (5*a^4*(a + b*x)^9)/(9*b^6) - (a^3*(a + b*x)^10)/b^
6 + (10*a^2*(a + b*x)^11)/(11*b^6) - (5*a*(a + b*x)^12)/(12*b^6) + (a + b*x)^13/
(13*b^6)

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Rubi in Sympy [A]  time = 16.9838, size = 90, normalized size = 0.94 \[ \frac{a^{7} x^{6}}{6} + a^{6} b x^{7} + \frac{21 a^{5} b^{2} x^{8}}{8} + \frac{35 a^{4} b^{3} x^{9}}{9} + \frac{7 a^{3} b^{4} x^{10}}{2} + \frac{21 a^{2} b^{5} x^{11}}{11} + \frac{7 a b^{6} x^{12}}{12} + \frac{b^{7} x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x+a)**7,x)

[Out]

a**7*x**6/6 + a**6*b*x**7 + 21*a**5*b**2*x**8/8 + 35*a**4*b**3*x**9/9 + 7*a**3*b
**4*x**10/2 + 21*a**2*b**5*x**11/11 + 7*a*b**6*x**12/12 + b**7*x**13/13

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Mathematica [A]  time = 0.00330446, size = 92, normalized size = 0.96 \[ \frac{a^7 x^6}{6}+a^6 b x^7+\frac{21}{8} a^5 b^2 x^8+\frac{35}{9} a^4 b^3 x^9+\frac{7}{2} a^3 b^4 x^{10}+\frac{21}{11} a^2 b^5 x^{11}+\frac{7}{12} a b^6 x^{12}+\frac{b^7 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x)^7,x]

[Out]

(a^7*x^6)/6 + a^6*b*x^7 + (21*a^5*b^2*x^8)/8 + (35*a^4*b^3*x^9)/9 + (7*a^3*b^4*x
^10)/2 + (21*a^2*b^5*x^11)/11 + (7*a*b^6*x^12)/12 + (b^7*x^13)/13

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Maple [A]  time = 0.004, size = 79, normalized size = 0.8 \[{\frac{{b}^{7}{x}^{13}}{13}}+{\frac{7\,a{b}^{6}{x}^{12}}{12}}+{\frac{21\,{a}^{2}{b}^{5}{x}^{11}}{11}}+{\frac{7\,{a}^{3}{b}^{4}{x}^{10}}{2}}+{\frac{35\,{a}^{4}{b}^{3}{x}^{9}}{9}}+{\frac{21\,{a}^{5}{b}^{2}{x}^{8}}{8}}+{a}^{6}b{x}^{7}+{\frac{{a}^{7}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x+a)^7,x)

[Out]

1/13*b^7*x^13+7/12*a*b^6*x^12+21/11*a^2*b^5*x^11+7/2*a^3*b^4*x^10+35/9*a^4*b^3*x
^9+21/8*a^5*b^2*x^8+a^6*b*x^7+1/6*a^7*x^6

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Maxima [A]  time = 1.34788, size = 105, normalized size = 1.09 \[ \frac{1}{13} \, b^{7} x^{13} + \frac{7}{12} \, a b^{6} x^{12} + \frac{21}{11} \, a^{2} b^{5} x^{11} + \frac{7}{2} \, a^{3} b^{4} x^{10} + \frac{35}{9} \, a^{4} b^{3} x^{9} + \frac{21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac{1}{6} \, a^{7} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7*x^5,x, algorithm="maxima")

[Out]

1/13*b^7*x^13 + 7/12*a*b^6*x^12 + 21/11*a^2*b^5*x^11 + 7/2*a^3*b^4*x^10 + 35/9*a
^4*b^3*x^9 + 21/8*a^5*b^2*x^8 + a^6*b*x^7 + 1/6*a^7*x^6

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Fricas [A]  time = 0.17936, size = 1, normalized size = 0.01 \[ \frac{1}{13} x^{13} b^{7} + \frac{7}{12} x^{12} b^{6} a + \frac{21}{11} x^{11} b^{5} a^{2} + \frac{7}{2} x^{10} b^{4} a^{3} + \frac{35}{9} x^{9} b^{3} a^{4} + \frac{21}{8} x^{8} b^{2} a^{5} + x^{7} b a^{6} + \frac{1}{6} x^{6} a^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7*x^5,x, algorithm="fricas")

[Out]

1/13*x^13*b^7 + 7/12*x^12*b^6*a + 21/11*x^11*b^5*a^2 + 7/2*x^10*b^4*a^3 + 35/9*x
^9*b^3*a^4 + 21/8*x^8*b^2*a^5 + x^7*b*a^6 + 1/6*x^6*a^7

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Sympy [A]  time = 0.147395, size = 90, normalized size = 0.94 \[ \frac{a^{7} x^{6}}{6} + a^{6} b x^{7} + \frac{21 a^{5} b^{2} x^{8}}{8} + \frac{35 a^{4} b^{3} x^{9}}{9} + \frac{7 a^{3} b^{4} x^{10}}{2} + \frac{21 a^{2} b^{5} x^{11}}{11} + \frac{7 a b^{6} x^{12}}{12} + \frac{b^{7} x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x+a)**7,x)

[Out]

a**7*x**6/6 + a**6*b*x**7 + 21*a**5*b**2*x**8/8 + 35*a**4*b**3*x**9/9 + 7*a**3*b
**4*x**10/2 + 21*a**2*b**5*x**11/11 + 7*a*b**6*x**12/12 + b**7*x**13/13

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GIAC/XCAS [A]  time = 0.206798, size = 105, normalized size = 1.09 \[ \frac{1}{13} \, b^{7} x^{13} + \frac{7}{12} \, a b^{6} x^{12} + \frac{21}{11} \, a^{2} b^{5} x^{11} + \frac{7}{2} \, a^{3} b^{4} x^{10} + \frac{35}{9} \, a^{4} b^{3} x^{9} + \frac{21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac{1}{6} \, a^{7} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7*x^5,x, algorithm="giac")

[Out]

1/13*b^7*x^13 + 7/12*a*b^6*x^12 + 21/11*a^2*b^5*x^11 + 7/2*a^3*b^4*x^10 + 35/9*a
^4*b^3*x^9 + 21/8*a^5*b^2*x^8 + a^6*b*x^7 + 1/6*a^7*x^6