3.2589 \(\int x^{-1-14 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=151 \[ -\frac{7 a^6 b^2 x^{-12 n}}{3 n}-\frac{56 a^5 b^3 x^{-11 n}}{11 n}-\frac{7 a^4 b^4 x^{-10 n}}{n}-\frac{56 a^3 b^5 x^{-9 n}}{9 n}-\frac{7 a^2 b^6 x^{-8 n}}{2 n}-\frac{8 a^7 b x^{-13 n}}{13 n}-\frac{a^8 x^{-14 n}}{14 n}-\frac{8 a b^7 x^{-7 n}}{7 n}-\frac{b^8 x^{-6 n}}{6 n} \]

[Out]

-a^8/(14*n*x^(14*n)) - (8*a^7*b)/(13*n*x^(13*n)) - (7*a^6*b^2)/(3*n*x^(12*n)) - (56*a^5*b^3)/(11*n*x^(11*n)) -
(7*a^4*b^4)/(n*x^(10*n)) - (56*a^3*b^5)/(9*n*x^(9*n)) - (7*a^2*b^6)/(2*n*x^(8*n)) - (8*a*b^7)/(7*n*x^(7*n)) -
b^8/(6*n*x^(6*n))

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Rubi [A]  time = 0.061078, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ -\frac{7 a^6 b^2 x^{-12 n}}{3 n}-\frac{56 a^5 b^3 x^{-11 n}}{11 n}-\frac{7 a^4 b^4 x^{-10 n}}{n}-\frac{56 a^3 b^5 x^{-9 n}}{9 n}-\frac{7 a^2 b^6 x^{-8 n}}{2 n}-\frac{8 a^7 b x^{-13 n}}{13 n}-\frac{a^8 x^{-14 n}}{14 n}-\frac{8 a b^7 x^{-7 n}}{7 n}-\frac{b^8 x^{-6 n}}{6 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 14*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(14*n*x^(14*n)) - (8*a^7*b)/(13*n*x^(13*n)) - (7*a^6*b^2)/(3*n*x^(12*n)) - (56*a^5*b^3)/(11*n*x^(11*n)) -
(7*a^4*b^4)/(n*x^(10*n)) - (56*a^3*b^5)/(9*n*x^(9*n)) - (7*a^2*b^6)/(2*n*x^(8*n)) - (8*a*b^7)/(7*n*x^(7*n)) -
b^8/(6*n*x^(6*n))

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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^{-1-14 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{15}} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^8}{x^{15}}+\frac{8 a^7 b}{x^{14}}+\frac{28 a^6 b^2}{x^{13}}+\frac{56 a^5 b^3}{x^{12}}+\frac{70 a^4 b^4}{x^{11}}+\frac{56 a^3 b^5}{x^{10}}+\frac{28 a^2 b^6}{x^9}+\frac{8 a b^7}{x^8}+\frac{b^8}{x^7}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-14 n}}{14 n}-\frac{8 a^7 b x^{-13 n}}{13 n}-\frac{7 a^6 b^2 x^{-12 n}}{3 n}-\frac{56 a^5 b^3 x^{-11 n}}{11 n}-\frac{7 a^4 b^4 x^{-10 n}}{n}-\frac{56 a^3 b^5 x^{-9 n}}{9 n}-\frac{7 a^2 b^6 x^{-8 n}}{2 n}-\frac{8 a b^7 x^{-7 n}}{7 n}-\frac{b^8 x^{-6 n}}{6 n}\\ \end{align*}

Mathematica [A]  time = 0.0506447, size = 113, normalized size = 0.75 \[ -\frac{x^{-14 n} \left (42042 a^6 b^2 x^{2 n}+91728 a^5 b^3 x^{3 n}+126126 a^4 b^4 x^{4 n}+112112 a^3 b^5 x^{5 n}+63063 a^2 b^6 x^{6 n}+11088 a^7 b x^n+1287 a^8+20592 a b^7 x^{7 n}+3003 b^8 x^{8 n}\right )}{18018 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 14*n)*(a + b*x^n)^8,x]

[Out]

-(1287*a^8 + 11088*a^7*b*x^n + 42042*a^6*b^2*x^(2*n) + 91728*a^5*b^3*x^(3*n) + 126126*a^4*b^4*x^(4*n) + 112112
*a^3*b^5*x^(5*n) + 63063*a^2*b^6*x^(6*n) + 20592*a*b^7*x^(7*n) + 3003*b^8*x^(8*n))/(18018*n*x^(14*n))

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Maple [A]  time = 0.025, size = 136, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{6\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{8\,{b}^{7}a}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{7\,{a}^{2}{b}^{6}}{2\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{56\,{a}^{3}{b}^{5}}{9\,n \left ({x}^{n} \right ) ^{9}}}-7\,{\frac{{a}^{4}{b}^{4}}{n \left ({x}^{n} \right ) ^{10}}}-{\frac{56\,{a}^{5}{b}^{3}}{11\,n \left ({x}^{n} \right ) ^{11}}}-{\frac{7\,{a}^{6}{b}^{2}}{3\,n \left ({x}^{n} \right ) ^{12}}}-{\frac{8\,b{a}^{7}}{13\,n \left ({x}^{n} \right ) ^{13}}}-{\frac{{a}^{8}}{14\,n \left ({x}^{n} \right ) ^{14}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-14*n)*(a+b*x^n)^8,x)

[Out]

-1/6*b^8/n/(x^n)^6-8/7*a*b^7/n/(x^n)^7-7/2*a^2*b^6/n/(x^n)^8-56/9*a^3*b^5/n/(x^n)^9-7*a^4*b^4/n/(x^n)^10-56/11
*a^5*b^3/n/(x^n)^11-7/3*a^6*b^2/n/(x^n)^12-8/13*a^7*b/n/(x^n)^13-1/14*a^8/n/(x^n)^14

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-14*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.10094, size = 290, normalized size = 1.92 \begin{align*} -\frac{3003 \, b^{8} x^{8 \, n} + 20592 \, a b^{7} x^{7 \, n} + 63063 \, a^{2} b^{6} x^{6 \, n} + 112112 \, a^{3} b^{5} x^{5 \, n} + 126126 \, a^{4} b^{4} x^{4 \, n} + 91728 \, a^{5} b^{3} x^{3 \, n} + 42042 \, a^{6} b^{2} x^{2 \, n} + 11088 \, a^{7} b x^{n} + 1287 \, a^{8}}{18018 \, n x^{14 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-14*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

-1/18018*(3003*b^8*x^(8*n) + 20592*a*b^7*x^(7*n) + 63063*a^2*b^6*x^(6*n) + 112112*a^3*b^5*x^(5*n) + 126126*a^4
*b^4*x^(4*n) + 91728*a^5*b^3*x^(3*n) + 42042*a^6*b^2*x^(2*n) + 11088*a^7*b*x^n + 1287*a^8)/(n*x^(14*n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-14*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.26223, size = 153, normalized size = 1.01 \begin{align*} -\frac{3003 \, b^{8} x^{8 \, n} + 20592 \, a b^{7} x^{7 \, n} + 63063 \, a^{2} b^{6} x^{6 \, n} + 112112 \, a^{3} b^{5} x^{5 \, n} + 126126 \, a^{4} b^{4} x^{4 \, n} + 91728 \, a^{5} b^{3} x^{3 \, n} + 42042 \, a^{6} b^{2} x^{2 \, n} + 11088 \, a^{7} b x^{n} + 1287 \, a^{8}}{18018 \, n x^{14 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-14*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

-1/18018*(3003*b^8*x^(8*n) + 20592*a*b^7*x^(7*n) + 63063*a^2*b^6*x^(6*n) + 112112*a^3*b^5*x^(5*n) + 126126*a^4
*b^4*x^(4*n) + 91728*a^5*b^3*x^(3*n) + 42042*a^6*b^2*x^(2*n) + 11088*a^7*b*x^n + 1287*a^8)/(n*x^(14*n))