3.128 \(\int x^6 (a+b x)^{10} \, dx\)

Optimal. Leaf size=112 \[ \frac{a^2 (a+b x)^{15}}{b^7}-\frac{10 a^3 (a+b x)^{14}}{7 b^7}+\frac{15 a^4 (a+b x)^{13}}{13 b^7}-\frac{a^5 (a+b x)^{12}}{2 b^7}+\frac{a^6 (a+b x)^{11}}{11 b^7}+\frac{(a+b x)^{17}}{17 b^7}-\frac{3 a (a+b x)^{16}}{8 b^7} \]

[Out]

(a^6*(a + b*x)^11)/(11*b^7) - (a^5*(a + b*x)^12)/(2*b^7) + (15*a^4*(a + b*x)^13)/(13*b^7) - (10*a^3*(a + b*x)^
14)/(7*b^7) + (a^2*(a + b*x)^15)/b^7 - (3*a*(a + b*x)^16)/(8*b^7) + (a + b*x)^17/(17*b^7)

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Rubi [A]  time = 0.0524065, antiderivative size = 112, 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} \[ \frac{a^2 (a+b x)^{15}}{b^7}-\frac{10 a^3 (a+b x)^{14}}{7 b^7}+\frac{15 a^4 (a+b x)^{13}}{13 b^7}-\frac{a^5 (a+b x)^{12}}{2 b^7}+\frac{a^6 (a+b x)^{11}}{11 b^7}+\frac{(a+b x)^{17}}{17 b^7}-\frac{3 a (a+b x)^{16}}{8 b^7} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x)^10,x]

[Out]

(a^6*(a + b*x)^11)/(11*b^7) - (a^5*(a + b*x)^12)/(2*b^7) + (15*a^4*(a + b*x)^13)/(13*b^7) - (10*a^3*(a + b*x)^
14)/(7*b^7) + (a^2*(a + b*x)^15)/b^7 - (3*a*(a + b*x)^16)/(8*b^7) + (a + b*x)^17/(17*b^7)

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^6 (a+b x)^{10} \, dx &=\int \left (\frac{a^6 (a+b x)^{10}}{b^6}-\frac{6 a^5 (a+b x)^{11}}{b^6}+\frac{15 a^4 (a+b x)^{12}}{b^6}-\frac{20 a^3 (a+b x)^{13}}{b^6}+\frac{15 a^2 (a+b x)^{14}}{b^6}-\frac{6 a (a+b x)^{15}}{b^6}+\frac{(a+b x)^{16}}{b^6}\right ) \, dx\\ &=\frac{a^6 (a+b x)^{11}}{11 b^7}-\frac{a^5 (a+b x)^{12}}{2 b^7}+\frac{15 a^4 (a+b x)^{13}}{13 b^7}-\frac{10 a^3 (a+b x)^{14}}{7 b^7}+\frac{a^2 (a+b x)^{15}}{b^7}-\frac{3 a (a+b x)^{16}}{8 b^7}+\frac{(a+b x)^{17}}{17 b^7}\\ \end{align*}

Mathematica [A]  time = 0.0074339, size = 126, normalized size = 1.12 \[ 3 a^2 b^8 x^{15}+\frac{60}{7} a^3 b^7 x^{14}+\frac{210}{13} a^4 b^6 x^{13}+21 a^5 b^5 x^{12}+\frac{210}{11} a^6 b^4 x^{11}+12 a^7 b^3 x^{10}+5 a^8 b^2 x^9+\frac{5}{4} a^9 b x^8+\frac{a^{10} x^7}{7}+\frac{5}{8} a b^9 x^{16}+\frac{b^{10} x^{17}}{17} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x)^10,x]

[Out]

(a^10*x^7)/7 + (5*a^9*b*x^8)/4 + 5*a^8*b^2*x^9 + 12*a^7*b^3*x^10 + (210*a^6*b^4*x^11)/11 + 21*a^5*b^5*x^12 + (
210*a^4*b^6*x^13)/13 + (60*a^3*b^7*x^14)/7 + 3*a^2*b^8*x^15 + (5*a*b^9*x^16)/8 + (b^10*x^17)/17

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Maple [A]  time = 0.001, size = 113, normalized size = 1. \begin{align*}{\frac{{b}^{10}{x}^{17}}{17}}+{\frac{5\,a{b}^{9}{x}^{16}}{8}}+3\,{a}^{2}{b}^{8}{x}^{15}+{\frac{60\,{a}^{3}{b}^{7}{x}^{14}}{7}}+{\frac{210\,{a}^{4}{b}^{6}{x}^{13}}{13}}+21\,{a}^{5}{b}^{5}{x}^{12}+{\frac{210\,{a}^{6}{b}^{4}{x}^{11}}{11}}+12\,{a}^{7}{b}^{3}{x}^{10}+5\,{a}^{8}{b}^{2}{x}^{9}+{\frac{5\,{a}^{9}b{x}^{8}}{4}}+{\frac{{a}^{10}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x+a)^10,x)

[Out]

1/17*b^10*x^17+5/8*a*b^9*x^16+3*a^2*b^8*x^15+60/7*a^3*b^7*x^14+210/13*a^4*b^6*x^13+21*a^5*b^5*x^12+210/11*a^6*
b^4*x^11+12*a^7*b^3*x^10+5*a^8*b^2*x^9+5/4*a^9*b*x^8+1/7*a^10*x^7

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Maxima [A]  time = 1.01775, size = 151, normalized size = 1.35 \begin{align*} \frac{1}{17} \, b^{10} x^{17} + \frac{5}{8} \, a b^{9} x^{16} + 3 \, a^{2} b^{8} x^{15} + \frac{60}{7} \, a^{3} b^{7} x^{14} + \frac{210}{13} \, a^{4} b^{6} x^{13} + 21 \, a^{5} b^{5} x^{12} + \frac{210}{11} \, a^{6} b^{4} x^{11} + 12 \, a^{7} b^{3} x^{10} + 5 \, a^{8} b^{2} x^{9} + \frac{5}{4} \, a^{9} b x^{8} + \frac{1}{7} \, a^{10} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^10,x, algorithm="maxima")

[Out]

1/17*b^10*x^17 + 5/8*a*b^9*x^16 + 3*a^2*b^8*x^15 + 60/7*a^3*b^7*x^14 + 210/13*a^4*b^6*x^13 + 21*a^5*b^5*x^12 +
210/11*a^6*b^4*x^11 + 12*a^7*b^3*x^10 + 5*a^8*b^2*x^9 + 5/4*a^9*b*x^8 + 1/7*a^10*x^7

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Fricas [A]  time = 1.53279, size = 266, normalized size = 2.38 \begin{align*} \frac{1}{17} x^{17} b^{10} + \frac{5}{8} x^{16} b^{9} a + 3 x^{15} b^{8} a^{2} + \frac{60}{7} x^{14} b^{7} a^{3} + \frac{210}{13} x^{13} b^{6} a^{4} + 21 x^{12} b^{5} a^{5} + \frac{210}{11} x^{11} b^{4} a^{6} + 12 x^{10} b^{3} a^{7} + 5 x^{9} b^{2} a^{8} + \frac{5}{4} x^{8} b a^{9} + \frac{1}{7} x^{7} a^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^10,x, algorithm="fricas")

[Out]

1/17*x^17*b^10 + 5/8*x^16*b^9*a + 3*x^15*b^8*a^2 + 60/7*x^14*b^7*a^3 + 210/13*x^13*b^6*a^4 + 21*x^12*b^5*a^5 +
210/11*x^11*b^4*a^6 + 12*x^10*b^3*a^7 + 5*x^9*b^2*a^8 + 5/4*x^8*b*a^9 + 1/7*x^7*a^10

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Sympy [A]  time = 0.106189, size = 128, normalized size = 1.14 \begin{align*} \frac{a^{10} x^{7}}{7} + \frac{5 a^{9} b x^{8}}{4} + 5 a^{8} b^{2} x^{9} + 12 a^{7} b^{3} x^{10} + \frac{210 a^{6} b^{4} x^{11}}{11} + 21 a^{5} b^{5} x^{12} + \frac{210 a^{4} b^{6} x^{13}}{13} + \frac{60 a^{3} b^{7} x^{14}}{7} + 3 a^{2} b^{8} x^{15} + \frac{5 a b^{9} x^{16}}{8} + \frac{b^{10} x^{17}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x+a)**10,x)

[Out]

a**10*x**7/7 + 5*a**9*b*x**8/4 + 5*a**8*b**2*x**9 + 12*a**7*b**3*x**10 + 210*a**6*b**4*x**11/11 + 21*a**5*b**5
*x**12 + 210*a**4*b**6*x**13/13 + 60*a**3*b**7*x**14/7 + 3*a**2*b**8*x**15 + 5*a*b**9*x**16/8 + b**10*x**17/17

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Giac [A]  time = 1.1522, size = 151, normalized size = 1.35 \begin{align*} \frac{1}{17} \, b^{10} x^{17} + \frac{5}{8} \, a b^{9} x^{16} + 3 \, a^{2} b^{8} x^{15} + \frac{60}{7} \, a^{3} b^{7} x^{14} + \frac{210}{13} \, a^{4} b^{6} x^{13} + 21 \, a^{5} b^{5} x^{12} + \frac{210}{11} \, a^{6} b^{4} x^{11} + 12 \, a^{7} b^{3} x^{10} + 5 \, a^{8} b^{2} x^{9} + \frac{5}{4} \, a^{9} b x^{8} + \frac{1}{7} \, a^{10} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^10,x, algorithm="giac")

[Out]

1/17*b^10*x^17 + 5/8*a*b^9*x^16 + 3*a^2*b^8*x^15 + 60/7*a^3*b^7*x^14 + 210/13*a^4*b^6*x^13 + 21*a^5*b^5*x^12 +
210/11*a^6*b^4*x^11 + 12*a^7*b^3*x^10 + 5*a^8*b^2*x^9 + 5/4*a^9*b*x^8 + 1/7*a^10*x^7