3.98 \(\int x^8 (a+b x)^7 \, dx\)

Optimal. Leaf size=95 \[ \frac{3}{2} a^2 b^5 x^{14}+\frac{35}{13} a^3 b^4 x^{13}+\frac{35}{12} a^4 b^3 x^{12}+\frac{21}{11} a^5 b^2 x^{11}+\frac{7}{10} a^6 b x^{10}+\frac{a^7 x^9}{9}+\frac{7}{15} a b^6 x^{15}+\frac{b^7 x^{16}}{16} \]

[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

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Rubi [A]  time = 0.0480825, 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, Rules used = {43} \[ \frac{3}{2} a^2 b^5 x^{14}+\frac{35}{13} a^3 b^4 x^{13}+\frac{35}{12} a^4 b^3 x^{12}+\frac{21}{11} a^5 b^2 x^{11}+\frac{7}{10} a^6 b x^{10}+\frac{a^7 x^9}{9}+\frac{7}{15} a b^6 x^{15}+\frac{b^7 x^{16}}{16} \]

Antiderivative was successfully verified.

[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 43

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*} \int x^8 (a+b x)^7 \, dx &=\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]  time = 0.0031677, size = 95, normalized size = 1. \[ \frac{3}{2} a^2 b^5 x^{14}+\frac{35}{13} a^3 b^4 x^{13}+\frac{35}{12} a^4 b^3 x^{12}+\frac{21}{11} a^5 b^2 x^{11}+\frac{7}{10} a^6 b x^{10}+\frac{a^7 x^9}{9}+\frac{7}{15} a b^6 x^{15}+\frac{b^7 x^{16}}{16} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.001, size = 80, normalized size = 0.8 \begin{align*}{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 1.01829, size = 107, normalized size = 1.13 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.38447, size = 198, normalized size = 2.08 \begin{align*} \frac{1}{16} x^{16} b^{7} + \frac{7}{15} x^{15} b^{6} a + \frac{3}{2} x^{14} b^{5} a^{2} + \frac{35}{13} x^{13} b^{4} a^{3} + \frac{35}{12} x^{12} b^{3} a^{4} + \frac{21}{11} x^{11} b^{2} a^{5} + \frac{7}{10} x^{10} b a^{6} + \frac{1}{9} x^{9} a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.09891, size = 94, normalized size = 0.99 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.18201, size = 107, normalized size = 1.13 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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