\(\int x^7 (a+b x)^7 \, dx\) [99]

Optimal result

Integrand size = 11, antiderivative size = 95 \[ \int x^7 (a+b x)^7 \, dx=\frac {a^7 x^8}{8}+\frac {7}{9} a^6 b x^9+\frac {21}{10} a^5 b^2 x^{10}+\frac {35}{11} a^4 b^3 x^{11}+\frac {35}{12} a^3 b^4 x^{12}+\frac {21}{13} a^2 b^5 x^{13}+\frac {1}{2} a b^6 x^{14}+\frac {b^7 x^{15}}{15} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 95, 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} \[ \int x^7 (a+b x)^7 \, dx=\frac {a^7 x^8}{8}+\frac {7}{9} a^6 b x^9+\frac {21}{10} a^5 b^2 x^{10}+\frac {35}{11} a^4 b^3 x^{11}+\frac {35}{12} a^3 b^4 x^{12}+\frac {21}{13} a^2 b^5 x^{13}+\frac {1}{2} a b^6 x^{14}+\frac {b^7 x^{15}}{15} \]

[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 + (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

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^7 x^7+7 a^6 b x^8+21 a^5 b^2 x^9+35 a^4 b^3 x^{10}+35 a^3 b^4 x^{11}+21 a^2 b^5 x^{12}+7 a b^6 x^{13}+b^7 x^{14}\right ) \, dx \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int x^7 (a+b x)^7 \, dx=\frac {a^7 x^8}{8}+\frac {7}{9} a^6 b x^9+\frac {21}{10} a^5 b^2 x^{10}+\frac {35}{11} a^4 b^3 x^{11}+\frac {35}{12} a^3 b^4 x^{12}+\frac {21}{13} a^2 b^5 x^{13}+\frac {1}{2} a b^6 x^{14}+\frac {b^7 x^{15}}{15} \]

[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 + (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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {1}{8} a^{7} x^{8}+\frac {7}{9} a^{6} b \,x^{9}+\frac {21}{10} a^{5} b^{2} x^{10}+\frac {35}{11} a^{4} b^{3} x^{11}+\frac {35}{12} a^{3} b^{4} x^{12}+\frac {21}{13} a^{2} b^{5} x^{13}+\frac {1}{2} a \,b^{6} x^{14}+\frac {1}{15} b^{7} x^{15}\)	\(80\)
default	\(\frac {1}{8} a^{7} x^{8}+\frac {7}{9} a^{6} b \,x^{9}+\frac {21}{10} a^{5} b^{2} x^{10}+\frac {35}{11} a^{4} b^{3} x^{11}+\frac {35}{12} a^{3} b^{4} x^{12}+\frac {21}{13} a^{2} b^{5} x^{13}+\frac {1}{2} a \,b^{6} x^{14}+\frac {1}{15} b^{7} x^{15}\)	\(80\)
norman	\(\frac {1}{8} a^{7} x^{8}+\frac {7}{9} a^{6} b \,x^{9}+\frac {21}{10} a^{5} b^{2} x^{10}+\frac {35}{11} a^{4} b^{3} x^{11}+\frac {35}{12} a^{3} b^{4} x^{12}+\frac {21}{13} a^{2} b^{5} x^{13}+\frac {1}{2} a \,b^{6} x^{14}+\frac {1}{15} b^{7} x^{15}\)	\(80\)
risch	\(\frac {1}{8} a^{7} x^{8}+\frac {7}{9} a^{6} b \,x^{9}+\frac {21}{10} a^{5} b^{2} x^{10}+\frac {35}{11} a^{4} b^{3} x^{11}+\frac {35}{12} a^{3} b^{4} x^{12}+\frac {21}{13} a^{2} b^{5} x^{13}+\frac {1}{2} a \,b^{6} x^{14}+\frac {1}{15} b^{7} x^{15}\)	\(80\)
parallelrisch	\(\frac {1}{8} a^{7} x^{8}+\frac {7}{9} a^{6} b \,x^{9}+\frac {21}{10} a^{5} b^{2} x^{10}+\frac {35}{11} a^{4} b^{3} x^{11}+\frac {35}{12} a^{3} b^{4} x^{12}+\frac {21}{13} a^{2} b^{5} x^{13}+\frac {1}{2} a \,b^{6} x^{14}+\frac {1}{15} b^{7} x^{15}\)	\(80\)

[In]

int(x^7*(b*x+a)^7,x,method=_RETURNVERBOSE)

[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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int x^7 (a+b x)^7 \, dx=\frac {1}{15} \, b^{7} x^{15} + \frac {1}{2} \, a b^{6} x^{14} + \frac {21}{13} \, a^{2} b^{5} x^{13} + \frac {35}{12} \, a^{3} b^{4} x^{12} + \frac {35}{11} \, a^{4} b^{3} x^{11} + \frac {21}{10} \, a^{5} b^{2} x^{10} + \frac {7}{9} \, a^{6} b x^{9} + \frac {1}{8} \, a^{7} x^{8} \]

[In]

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

[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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.97 \[ \int x^7 (a+b x)^7 \, dx=\frac {a^{7} x^{8}}{8} + \frac {7 a^{6} b x^{9}}{9} + \frac {21 a^{5} b^{2} x^{10}}{10} + \frac {35 a^{4} b^{3} x^{11}}{11} + \frac {35 a^{3} b^{4} x^{12}}{12} + \frac {21 a^{2} b^{5} x^{13}}{13} + \frac {a b^{6} x^{14}}{2} + \frac {b^{7} x^{15}}{15} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int x^7 (a+b x)^7 \, dx=\frac {1}{15} \, b^{7} x^{15} + \frac {1}{2} \, a b^{6} x^{14} + \frac {21}{13} \, a^{2} b^{5} x^{13} + \frac {35}{12} \, a^{3} b^{4} x^{12} + \frac {35}{11} \, a^{4} b^{3} x^{11} + \frac {21}{10} \, a^{5} b^{2} x^{10} + \frac {7}{9} \, a^{6} b x^{9} + \frac {1}{8} \, a^{7} x^{8} \]

[In]

integrate(x^7*(b*x+a)^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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int x^7 (a+b x)^7 \, dx=\frac {1}{15} \, b^{7} x^{15} + \frac {1}{2} \, a b^{6} x^{14} + \frac {21}{13} \, a^{2} b^{5} x^{13} + \frac {35}{12} \, a^{3} b^{4} x^{12} + \frac {35}{11} \, a^{4} b^{3} x^{11} + \frac {21}{10} \, a^{5} b^{2} x^{10} + \frac {7}{9} \, a^{6} b x^{9} + \frac {1}{8} \, a^{7} x^{8} \]

[In]

integrate(x^7*(b*x+a)^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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int x^7 (a+b x)^7 \, dx=\frac {a^7\,x^8}{8}+\frac {7\,a^6\,b\,x^9}{9}+\frac {21\,a^5\,b^2\,x^{10}}{10}+\frac {35\,a^4\,b^3\,x^{11}}{11}+\frac {35\,a^3\,b^4\,x^{12}}{12}+\frac {21\,a^2\,b^5\,x^{13}}{13}+\frac {a\,b^6\,x^{14}}{2}+\frac {b^7\,x^{15}}{15} \]

[In]

int(x^7*(a + b*x)^7,x)

[Out]

(a^7*x^8)/8 + (b^7*x^15)/15 + (7*a^6*b*x^9)/9 + (a*b^6*x^14)/2 + (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