\(\int x^6 (a+b x)^5 \, dx\) [77]

Optimal result

Integrand size = 11, antiderivative size = 66 \[ \int x^6 (a+b x)^5 \, dx=\frac {a^5 x^7}{7}+\frac {5}{8} a^4 b x^8+\frac {10}{9} a^3 b^2 x^9+a^2 b^3 x^{10}+\frac {5}{11} a b^4 x^{11}+\frac {b^5 x^{12}}{12} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x^6 (a+b x)^5 \, dx=\frac {a^5 x^7}{7}+\frac {5}{8} a^4 b x^8+\frac {10}{9} a^3 b^2 x^9+a^2 b^3 x^{10}+\frac {5}{11} a b^4 x^{11}+\frac {b^5 x^{12}}{12} \]

[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^11)/11 + (b^5*x^12)/12

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 x^6+5 a^4 b x^7+10 a^3 b^2 x^8+10 a^2 b^3 x^9+5 a b^4 x^{10}+b^5 x^{11}\right ) \, dx \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int x^6 (a+b x)^5 \, dx=\frac {a^5 x^7}{7}+\frac {5}{8} a^4 b x^8+\frac {10}{9} a^3 b^2 x^9+a^2 b^3 x^{10}+\frac {5}{11} a b^4 x^{11}+\frac {b^5 x^{12}}{12} \]

[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^11)/11 + (b^5*x^12)/12

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86

method	result	size
gosper	\(\frac {1}{7} a^{5} x^{7}+\frac {5}{8} a^{4} b \,x^{8}+\frac {10}{9} a^{3} b^{2} x^{9}+a^{2} b^{3} x^{10}+\frac {5}{11} a \,b^{4} x^{11}+\frac {1}{12} b^{5} x^{12}\)	\(57\)
default	\(\frac {1}{7} a^{5} x^{7}+\frac {5}{8} a^{4} b \,x^{8}+\frac {10}{9} a^{3} b^{2} x^{9}+a^{2} b^{3} x^{10}+\frac {5}{11} a \,b^{4} x^{11}+\frac {1}{12} b^{5} x^{12}\)	\(57\)
norman	\(\frac {1}{7} a^{5} x^{7}+\frac {5}{8} a^{4} b \,x^{8}+\frac {10}{9} a^{3} b^{2} x^{9}+a^{2} b^{3} x^{10}+\frac {5}{11} a \,b^{4} x^{11}+\frac {1}{12} b^{5} x^{12}\)	\(57\)
risch	\(\frac {1}{7} a^{5} x^{7}+\frac {5}{8} a^{4} b \,x^{8}+\frac {10}{9} a^{3} b^{2} x^{9}+a^{2} b^{3} x^{10}+\frac {5}{11} a \,b^{4} x^{11}+\frac {1}{12} b^{5} x^{12}\)	\(57\)
parallelrisch	\(\frac {1}{7} a^{5} x^{7}+\frac {5}{8} a^{4} b \,x^{8}+\frac {10}{9} a^{3} b^{2} x^{9}+a^{2} b^{3} x^{10}+\frac {5}{11} a \,b^{4} x^{11}+\frac {1}{12} b^{5} x^{12}\)	\(57\)

[In]

int(x^6*(b*x+a)^5,x,method=_RETURNVERBOSE)

[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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int x^6 (a+b x)^5 \, dx=\frac {1}{12} \, b^{5} x^{12} + \frac {5}{11} \, a b^{4} x^{11} + a^{2} b^{3} x^{10} + \frac {10}{9} \, a^{3} b^{2} x^{9} + \frac {5}{8} \, a^{4} b x^{8} + \frac {1}{7} \, a^{5} x^{7} \]

[In]

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

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int x^6 (a+b x)^5 \, dx=\frac {a^{5} x^{7}}{7} + \frac {5 a^{4} b x^{8}}{8} + \frac {10 a^{3} b^{2} x^{9}}{9} + a^{2} b^{3} x^{10} + \frac {5 a b^{4} x^{11}}{11} + \frac {b^{5} x^{12}}{12} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int x^6 (a+b x)^5 \, dx=\frac {1}{12} \, b^{5} x^{12} + \frac {5}{11} \, a b^{4} x^{11} + a^{2} b^{3} x^{10} + \frac {10}{9} \, a^{3} b^{2} x^{9} + \frac {5}{8} \, a^{4} b x^{8} + \frac {1}{7} \, a^{5} x^{7} \]

[In]

integrate(x^6*(b*x+a)^5,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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int x^6 (a+b x)^5 \, dx=\frac {1}{12} \, b^{5} x^{12} + \frac {5}{11} \, a b^{4} x^{11} + a^{2} b^{3} x^{10} + \frac {10}{9} \, a^{3} b^{2} x^{9} + \frac {5}{8} \, a^{4} b x^{8} + \frac {1}{7} \, a^{5} x^{7} \]

[In]

integrate(x^6*(b*x+a)^5,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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int x^6 (a+b x)^5 \, dx=\frac {a^5\,x^7}{7}+\frac {5\,a^4\,b\,x^8}{8}+\frac {10\,a^3\,b^2\,x^9}{9}+a^2\,b^3\,x^{10}+\frac {5\,a\,b^4\,x^{11}}{11}+\frac {b^5\,x^{12}}{12} \]

[In]

int(x^6*(a + b*x)^5,x)

[Out]

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