3.77 \(\int x^6 (a+b x)^5 \, dx\)

Optimal. Leaf size=66 \[ \frac{a^5 x^7}{7}+\frac{5}{8} a^4 b x^8+\frac{10}{9} a^3 b^2 x^9+a^2 b^3 x^{10}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{12}}{12} \]

[Out]

(a^5*x^7)/7 + (5*a^4*b*x^8)/8 + (10*a^3*b^2*x^9)/9 + a^2*b^3*x^10 + (5*a*b^4*x^1
1)/11 + (b^5*x^12)/12

_______________________________________________________________________________________

Rubi [A]  time = 0.0728336, antiderivative size = 66, 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 \[ \frac{a^5 x^7}{7}+\frac{5}{8} a^4 b x^8+\frac{10}{9} a^3 b^2 x^9+a^2 b^3 x^{10}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a + b*x)^5,x]

[Out]

(a^5*x^7)/7 + (5*a^4*b*x^8)/8 + (10*a^3*b^2*x^9)/9 + a^2*b^3*x^10 + (5*a*b^4*x^1
1)/11 + (b^5*x^12)/12

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.3958, size = 63, normalized size = 0.95 \[ \frac{a^{5} x^{7}}{7} + \frac{5 a^{4} b x^{8}}{8} + \frac{10 a^{3} b^{2} x^{9}}{9} + a^{2} b^{3} x^{10} + \frac{5 a b^{4} x^{11}}{11} + \frac{b^{5} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(b*x+a)**5,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00343054, size = 66, normalized size = 1. \[ \frac{a^5 x^7}{7}+\frac{5}{8} a^4 b x^8+\frac{10}{9} a^3 b^2 x^9+a^2 b^3 x^{10}+\frac{5}{11} a b^4 x^{11}+\frac{b^5 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6*(a + b*x)^5,x]

[Out]

(a^5*x^7)/7 + (5*a^4*b*x^8)/8 + (10*a^3*b^2*x^9)/9 + a^2*b^3*x^10 + (5*a*b^4*x^1
1)/11 + (b^5*x^12)/12

_______________________________________________________________________________________

Maple [A]  time = 0.036, size = 57, normalized size = 0.9 \[{\frac{{a}^{5}{x}^{7}}{7}}+{\frac{5\,{a}^{4}b{x}^{8}}{8}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{9}}{9}}+{a}^{2}{b}^{3}{x}^{10}+{\frac{5\,a{b}^{4}{x}^{11}}{11}}+{\frac{{b}^{5}{x}^{12}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(b*x+a)^5,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.45358, size = 76, normalized size = 1.15 \[ \frac{1}{12} \, b^{5} x^{12} + \frac{5}{11} \, a b^{4} x^{11} + a^{2} b^{3} x^{10} + \frac{10}{9} \, a^{3} b^{2} x^{9} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{7} \, a^{5} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*x^6,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.181033, size = 1, normalized size = 0.02 \[ \frac{1}{12} x^{12} b^{5} + \frac{5}{11} x^{11} b^{4} a + x^{10} b^{3} a^{2} + \frac{10}{9} x^{9} b^{2} a^{3} + \frac{5}{8} x^{8} b a^{4} + \frac{1}{7} x^{7} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*x^6,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.122087, size = 63, normalized size = 0.95 \[ \frac{a^{5} x^{7}}{7} + \frac{5 a^{4} b x^{8}}{8} + \frac{10 a^{3} b^{2} x^{9}}{9} + a^{2} b^{3} x^{10} + \frac{5 a b^{4} x^{11}}{11} + \frac{b^{5} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(b*x+a)**5,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21146, size = 76, normalized size = 1.15 \[ \frac{1}{12} \, b^{5} x^{12} + \frac{5}{11} \, a b^{4} x^{11} + a^{2} b^{3} x^{10} + \frac{10}{9} \, a^{3} b^{2} x^{9} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{7} \, a^{5} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5*x^6,x, algorithm="giac")

[Out]

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