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

Optimal. Leaf size=95 \[ \frac{a^7 x^8}{8}+\frac{7}{9} a^6 b x^9+\frac{21}{10} a^5 b^2 x^{10}+\frac{35}{11} a^4 b^3 x^{11}+\frac{35}{12} a^3 b^4 x^{12}+\frac{21}{13} a^2 b^5 x^{13}+\frac{1}{2} a b^6 x^{14}+\frac{b^7 x^{15}}{15} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0912505, antiderivative size = 95, 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 \[ \frac{a^7 x^8}{8}+\frac{7}{9} a^6 b x^9+\frac{21}{10} a^5 b^2 x^{10}+\frac{35}{11} a^4 b^3 x^{11}+\frac{35}{12} a^3 b^4 x^{12}+\frac{21}{13} a^2 b^5 x^{13}+\frac{1}{2} a b^6 x^{14}+\frac{b^7 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + b*x)^7,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.8805, size = 92, normalized size = 0.97 \[ \frac{a^{7} x^{8}}{8} + \frac{7 a^{6} b x^{9}}{9} + \frac{21 a^{5} b^{2} x^{10}}{10} + \frac{35 a^{4} b^{3} x^{11}}{11} + \frac{35 a^{3} b^{4} x^{12}}{12} + \frac{21 a^{2} b^{5} x^{13}}{13} + \frac{a b^{6} x^{14}}{2} + \frac{b^{7} x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00351693, size = 95, normalized size = 1. \[ \frac{a^7 x^8}{8}+\frac{7}{9} a^6 b x^9+\frac{21}{10} a^5 b^2 x^{10}+\frac{35}{11} a^4 b^3 x^{11}+\frac{35}{12} a^3 b^4 x^{12}+\frac{21}{13} a^2 b^5 x^{13}+\frac{1}{2} a b^6 x^{14}+\frac{b^7 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a + b*x)^7,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 80, normalized size = 0.8 \[{\frac{{a}^{7}{x}^{8}}{8}}+{\frac{7\,{a}^{6}b{x}^{9}}{9}}+{\frac{21\,{a}^{5}{b}^{2}{x}^{10}}{10}}+{\frac{35\,{a}^{4}{b}^{3}{x}^{11}}{11}}+{\frac{35\,{a}^{3}{b}^{4}{x}^{12}}{12}}+{\frac{21\,{a}^{2}{b}^{5}{x}^{13}}{13}}+{\frac{a{b}^{6}{x}^{14}}{2}}+{\frac{{b}^{7}{x}^{15}}{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x+a)^7,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34719, size = 107, normalized size = 1.13 \[ \frac{1}{15} \, b^{7} x^{15} + \frac{1}{2} \, a b^{6} x^{14} + \frac{21}{13} \, a^{2} b^{5} x^{13} + \frac{35}{12} \, a^{3} b^{4} x^{12} + \frac{35}{11} \, a^{4} b^{3} x^{11} + \frac{21}{10} \, a^{5} b^{2} x^{10} + \frac{7}{9} \, a^{6} b x^{9} + \frac{1}{8} \, a^{7} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.174687, size = 1, normalized size = 0.01 \[ \frac{1}{15} x^{15} b^{7} + \frac{1}{2} x^{14} b^{6} a + \frac{21}{13} x^{13} b^{5} a^{2} + \frac{35}{12} x^{12} b^{4} a^{3} + \frac{35}{11} x^{11} b^{3} a^{4} + \frac{21}{10} x^{10} b^{2} a^{5} + \frac{7}{9} x^{9} b a^{6} + \frac{1}{8} x^{8} a^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.143411, size = 92, normalized size = 0.97 \[ \frac{a^{7} x^{8}}{8} + \frac{7 a^{6} b x^{9}}{9} + \frac{21 a^{5} b^{2} x^{10}}{10} + \frac{35 a^{4} b^{3} x^{11}}{11} + \frac{35 a^{3} b^{4} x^{12}}{12} + \frac{21 a^{2} b^{5} x^{13}}{13} + \frac{a b^{6} x^{14}}{2} + \frac{b^{7} x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b*x+a)**7,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216858, size = 107, normalized size = 1.13 \[ \frac{1}{15} \, b^{7} x^{15} + \frac{1}{2} \, a b^{6} x^{14} + \frac{21}{13} \, a^{2} b^{5} x^{13} + \frac{35}{12} \, a^{3} b^{4} x^{12} + \frac{35}{11} \, a^{4} b^{3} x^{11} + \frac{21}{10} \, a^{5} b^{2} x^{10} + \frac{7}{9} \, a^{6} b x^{9} + \frac{1}{8} \, a^{7} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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