\(\int x^8 (a+b x)^7 \, dx\) [98]

Optimal result

Integrand size = 11, antiderivative size = 95 \[ \int x^8 (a+b x)^7 \, dx=\frac {a^7 x^9}{9}+\frac {7}{10} a^6 b x^{10}+\frac {21}{11} a^5 b^2 x^{11}+\frac {35}{12} a^4 b^3 x^{12}+\frac {35}{13} a^3 b^4 x^{13}+\frac {3}{2} a^2 b^5 x^{14}+\frac {7}{15} a b^6 x^{15}+\frac {b^7 x^{16}}{16} \]

[Out]

1/9*a^7*x^9+7/10*a^6*b*x^10+21/11*a^5*b^2*x^11+35/12*a^4*b^3*x^12+35/13*a^3*b^4*x^13+3/2*a^2*b^5*x^14+7/15*a*b
^6*x^15+1/16*b^7*x^16

Rubi [A] (verified)

Time = 0.03 (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} \[ \int x^8 (a+b x)^7 \, dx=\frac {a^7 x^9}{9}+\frac {7}{10} a^6 b x^{10}+\frac {21}{11} a^5 b^2 x^{11}+\frac {35}{12} a^4 b^3 x^{12}+\frac {35}{13} a^3 b^4 x^{13}+\frac {3}{2} a^2 b^5 x^{14}+\frac {7}{15} a b^6 x^{15}+\frac {b^7 x^{16}}{16} \]

[In]

Int[x^8*(a + b*x)^7,x]

[Out]

(a^7*x^9)/9 + (7*a^6*b*x^10)/10 + (21*a^5*b^2*x^11)/11 + (35*a^4*b^3*x^12)/12 + (35*a^3*b^4*x^13)/13 + (3*a^2*
b^5*x^14)/2 + (7*a*b^6*x^15)/15 + (b^7*x^16)/16

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^8+7 a^6 b x^9+21 a^5 b^2 x^{10}+35 a^4 b^3 x^{11}+35 a^3 b^4 x^{12}+21 a^2 b^5 x^{13}+7 a b^6 x^{14}+b^7 x^{15}\right ) \, dx \\ & = \frac {a^7 x^9}{9}+\frac {7}{10} a^6 b x^{10}+\frac {21}{11} a^5 b^2 x^{11}+\frac {35}{12} a^4 b^3 x^{12}+\frac {35}{13} a^3 b^4 x^{13}+\frac {3}{2} a^2 b^5 x^{14}+\frac {7}{15} a b^6 x^{15}+\frac {b^7 x^{16}}{16} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int x^8 (a+b x)^7 \, dx=\frac {a^7 x^9}{9}+\frac {7}{10} a^6 b x^{10}+\frac {21}{11} a^5 b^2 x^{11}+\frac {35}{12} a^4 b^3 x^{12}+\frac {35}{13} a^3 b^4 x^{13}+\frac {3}{2} a^2 b^5 x^{14}+\frac {7}{15} a b^6 x^{15}+\frac {b^7 x^{16}}{16} \]

[In]

Integrate[x^8*(a + b*x)^7,x]

[Out]

(a^7*x^9)/9 + (7*a^6*b*x^10)/10 + (21*a^5*b^2*x^11)/11 + (35*a^4*b^3*x^12)/12 + (35*a^3*b^4*x^13)/13 + (3*a^2*
b^5*x^14)/2 + (7*a*b^6*x^15)/15 + (b^7*x^16)/16

Maple [A] (verified)

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

method	result	size
gosper	\(\frac {1}{9} a^{7} x^{9}+\frac {7}{10} a^{6} b \,x^{10}+\frac {21}{11} a^{5} b^{2} x^{11}+\frac {35}{12} a^{4} b^{3} x^{12}+\frac {35}{13} a^{3} b^{4} x^{13}+\frac {3}{2} a^{2} b^{5} x^{14}+\frac {7}{15} a \,b^{6} x^{15}+\frac {1}{16} b^{7} x^{16}\)	\(80\)
default	\(\frac {1}{9} a^{7} x^{9}+\frac {7}{10} a^{6} b \,x^{10}+\frac {21}{11} a^{5} b^{2} x^{11}+\frac {35}{12} a^{4} b^{3} x^{12}+\frac {35}{13} a^{3} b^{4} x^{13}+\frac {3}{2} a^{2} b^{5} x^{14}+\frac {7}{15} a \,b^{6} x^{15}+\frac {1}{16} b^{7} x^{16}\)	\(80\)
norman	\(\frac {1}{9} a^{7} x^{9}+\frac {7}{10} a^{6} b \,x^{10}+\frac {21}{11} a^{5} b^{2} x^{11}+\frac {35}{12} a^{4} b^{3} x^{12}+\frac {35}{13} a^{3} b^{4} x^{13}+\frac {3}{2} a^{2} b^{5} x^{14}+\frac {7}{15} a \,b^{6} x^{15}+\frac {1}{16} b^{7} x^{16}\)	\(80\)
risch	\(\frac {1}{9} a^{7} x^{9}+\frac {7}{10} a^{6} b \,x^{10}+\frac {21}{11} a^{5} b^{2} x^{11}+\frac {35}{12} a^{4} b^{3} x^{12}+\frac {35}{13} a^{3} b^{4} x^{13}+\frac {3}{2} a^{2} b^{5} x^{14}+\frac {7}{15} a \,b^{6} x^{15}+\frac {1}{16} b^{7} x^{16}\)	\(80\)
parallelrisch	\(\frac {1}{9} a^{7} x^{9}+\frac {7}{10} a^{6} b \,x^{10}+\frac {21}{11} a^{5} b^{2} x^{11}+\frac {35}{12} a^{4} b^{3} x^{12}+\frac {35}{13} a^{3} b^{4} x^{13}+\frac {3}{2} a^{2} b^{5} x^{14}+\frac {7}{15} a \,b^{6} x^{15}+\frac {1}{16} b^{7} x^{16}\)	\(80\)

[In]

int(x^8*(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

1/9*a^7*x^9+7/10*a^6*b*x^10+21/11*a^5*b^2*x^11+35/12*a^4*b^3*x^12+35/13*a^3*b^4*x^13+3/2*a^2*b^5*x^14+7/15*a*b
^6*x^15+1/16*b^7*x^16

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int x^8 (a+b x)^7 \, dx=\frac {1}{16} \, b^{7} x^{16} + \frac {7}{15} \, a b^{6} x^{15} + \frac {3}{2} \, a^{2} b^{5} x^{14} + \frac {35}{13} \, a^{3} b^{4} x^{13} + \frac {35}{12} \, a^{4} b^{3} x^{12} + \frac {21}{11} \, a^{5} b^{2} x^{11} + \frac {7}{10} \, a^{6} b x^{10} + \frac {1}{9} \, a^{7} x^{9} \]

[In]

integrate(x^8*(b*x+a)^7,x, algorithm="fricas")

[Out]

1/16*b^7*x^16 + 7/15*a*b^6*x^15 + 3/2*a^2*b^5*x^14 + 35/13*a^3*b^4*x^13 + 35/12*a^4*b^3*x^12 + 21/11*a^5*b^2*x
^11 + 7/10*a^6*b*x^10 + 1/9*a^7*x^9

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.99 \[ \int x^8 (a+b x)^7 \, dx=\frac {a^{7} x^{9}}{9} + \frac {7 a^{6} b x^{10}}{10} + \frac {21 a^{5} b^{2} x^{11}}{11} + \frac {35 a^{4} b^{3} x^{12}}{12} + \frac {35 a^{3} b^{4} x^{13}}{13} + \frac {3 a^{2} b^{5} x^{14}}{2} + \frac {7 a b^{6} x^{15}}{15} + \frac {b^{7} x^{16}}{16} \]

[In]

integrate(x**8*(b*x+a)**7,x)

[Out]

a**7*x**9/9 + 7*a**6*b*x**10/10 + 21*a**5*b**2*x**11/11 + 35*a**4*b**3*x**12/12 + 35*a**3*b**4*x**13/13 + 3*a*
*2*b**5*x**14/2 + 7*a*b**6*x**15/15 + b**7*x**16/16

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int x^8 (a+b x)^7 \, dx=\frac {1}{16} \, b^{7} x^{16} + \frac {7}{15} \, a b^{6} x^{15} + \frac {3}{2} \, a^{2} b^{5} x^{14} + \frac {35}{13} \, a^{3} b^{4} x^{13} + \frac {35}{12} \, a^{4} b^{3} x^{12} + \frac {21}{11} \, a^{5} b^{2} x^{11} + \frac {7}{10} \, a^{6} b x^{10} + \frac {1}{9} \, a^{7} x^{9} \]

[In]

integrate(x^8*(b*x+a)^7,x, algorithm="maxima")

[Out]

1/16*b^7*x^16 + 7/15*a*b^6*x^15 + 3/2*a^2*b^5*x^14 + 35/13*a^3*b^4*x^13 + 35/12*a^4*b^3*x^12 + 21/11*a^5*b^2*x
^11 + 7/10*a^6*b*x^10 + 1/9*a^7*x^9

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int x^8 (a+b x)^7 \, dx=\frac {1}{16} \, b^{7} x^{16} + \frac {7}{15} \, a b^{6} x^{15} + \frac {3}{2} \, a^{2} b^{5} x^{14} + \frac {35}{13} \, a^{3} b^{4} x^{13} + \frac {35}{12} \, a^{4} b^{3} x^{12} + \frac {21}{11} \, a^{5} b^{2} x^{11} + \frac {7}{10} \, a^{6} b x^{10} + \frac {1}{9} \, a^{7} x^{9} \]

[In]

integrate(x^8*(b*x+a)^7,x, algorithm="giac")

[Out]

1/16*b^7*x^16 + 7/15*a*b^6*x^15 + 3/2*a^2*b^5*x^14 + 35/13*a^3*b^4*x^13 + 35/12*a^4*b^3*x^12 + 21/11*a^5*b^2*x
^11 + 7/10*a^6*b*x^10 + 1/9*a^7*x^9

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.83 \[ \int x^8 (a+b x)^7 \, dx=\frac {a^7\,x^9}{9}+\frac {7\,a^6\,b\,x^{10}}{10}+\frac {21\,a^5\,b^2\,x^{11}}{11}+\frac {35\,a^4\,b^3\,x^{12}}{12}+\frac {35\,a^3\,b^4\,x^{13}}{13}+\frac {3\,a^2\,b^5\,x^{14}}{2}+\frac {7\,a\,b^6\,x^{15}}{15}+\frac {b^7\,x^{16}}{16} \]

[In]

int(x^8*(a + b*x)^7,x)

[Out]

(a^7*x^9)/9 + (b^7*x^16)/16 + (7*a^6*b*x^10)/10 + (7*a*b^6*x^15)/15 + (21*a^5*b^2*x^11)/11 + (35*a^4*b^3*x^12)
/12 + (35*a^3*b^4*x^13)/13 + (3*a^2*b^5*x^14)/2