3.2588 \(\int x^{-1-13 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=131 \[ -\frac{b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}+\frac{b^3 x^{-10 n} \left (a+b x^n\right )^9}{715 a^4 n}-\frac{b^4 x^{-9 n} \left (a+b x^n\right )^9}{6435 a^5 n}+\frac{b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac{x^{-13 n} \left (a+b x^n\right )^9}{13 a n} \]

[Out]

-(a + b*x^n)^9/(13*a*n*x^(13*n)) + (b*(a + b*x^n)^9)/(39*a^2*n*x^(12*n)) - (b^2*(a + b*x^n)^9)/(143*a^3*n*x^(1
1*n)) + (b^3*(a + b*x^n)^9)/(715*a^4*n*x^(10*n)) - (b^4*(a + b*x^n)^9)/(6435*a^5*n*x^(9*n))

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Rubi [A]  time = 0.0538214, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {266, 45, 37} \[ -\frac{b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}+\frac{b^3 x^{-10 n} \left (a+b x^n\right )^9}{715 a^4 n}-\frac{b^4 x^{-9 n} \left (a+b x^n\right )^9}{6435 a^5 n}+\frac{b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac{x^{-13 n} \left (a+b x^n\right )^9}{13 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 13*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(13*a*n*x^(13*n)) + (b*(a + b*x^n)^9)/(39*a^2*n*x^(12*n)) - (b^2*(a + b*x^n)^9)/(143*a^3*n*x^(1
1*n)) + (b^3*(a + b*x^n)^9)/(715*a^4*n*x^(10*n)) - (b^4*(a + b*x^n)^9)/(6435*a^5*n*x^(9*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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int x^{-1-13 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{14}} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-13 n} \left (a+b x^n\right )^9}{13 a n}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{13}} \, dx,x,x^n\right )}{13 a n}\\ &=-\frac{x^{-13 n} \left (a+b x^n\right )^9}{13 a n}+\frac{b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{12}} \, dx,x,x^n\right )}{13 a^2 n}\\ &=-\frac{x^{-13 n} \left (a+b x^n\right )^9}{13 a n}+\frac{b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac{b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^n\right )}{143 a^3 n}\\ &=-\frac{x^{-13 n} \left (a+b x^n\right )^9}{13 a n}+\frac{b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac{b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}+\frac{b^3 x^{-10 n} \left (a+b x^n\right )^9}{715 a^4 n}+\frac{b^4 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^n\right )}{715 a^4 n}\\ &=-\frac{x^{-13 n} \left (a+b x^n\right )^9}{13 a n}+\frac{b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac{b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}+\frac{b^3 x^{-10 n} \left (a+b x^n\right )^9}{715 a^4 n}-\frac{b^4 x^{-9 n} \left (a+b x^n\right )^9}{6435 a^5 n}\\ \end{align*}

Mathematica [A]  time = 0.0503247, size = 113, normalized size = 0.86 \[ -\frac{x^{-13 n} \left (16380 a^6 b^2 x^{2 n}+36036 a^5 b^3 x^{3 n}+50050 a^4 b^4 x^{4 n}+45045 a^3 b^5 x^{5 n}+25740 a^2 b^6 x^{6 n}+4290 a^7 b x^n+495 a^8+8580 a b^7 x^{7 n}+1287 b^8 x^{8 n}\right )}{6435 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 13*n)*(a + b*x^n)^8,x]

[Out]

-(495*a^8 + 4290*a^7*b*x^n + 16380*a^6*b^2*x^(2*n) + 36036*a^5*b^3*x^(3*n) + 50050*a^4*b^4*x^(4*n) + 45045*a^3
*b^5*x^(5*n) + 25740*a^2*b^6*x^(6*n) + 8580*a*b^7*x^(7*n) + 1287*b^8*x^(8*n))/(6435*n*x^(13*n))

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Maple [A]  time = 0.023, size = 136, normalized size = 1. \begin{align*} -{\frac{{b}^{8}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{4\,{b}^{7}a}{3\,n \left ({x}^{n} \right ) ^{6}}}-4\,{\frac{{a}^{2}{b}^{6}}{n \left ({x}^{n} \right ) ^{7}}}-7\,{\frac{{a}^{3}{b}^{5}}{n \left ({x}^{n} \right ) ^{8}}}-{\frac{70\,{a}^{4}{b}^{4}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{28\,{a}^{5}{b}^{3}}{5\,n \left ({x}^{n} \right ) ^{10}}}-{\frac{28\,{a}^{6}{b}^{2}}{11\,n \left ({x}^{n} \right ) ^{11}}}-{\frac{2\,b{a}^{7}}{3\,n \left ({x}^{n} \right ) ^{12}}}-{\frac{{a}^{8}}{13\,n \left ({x}^{n} \right ) ^{13}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-13*n)*(a+b*x^n)^8,x)

[Out]

-1/5*b^8/n/(x^n)^5-4/3*a*b^7/n/(x^n)^6-4*a^2*b^6/n/(x^n)^7-7*a^3*b^5/n/(x^n)^8-70/9*a^4*b^4/n/(x^n)^9-28/5*a^5
*b^3/n/(x^n)^10-28/11*a^6*b^2/n/(x^n)^11-2/3*a^7*b/n/(x^n)^12-1/13*a^8/n/(x^n)^13

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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^(-1-13*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.01941, size = 282, normalized size = 2.15 \begin{align*} -\frac{1287 \, b^{8} x^{8 \, n} + 8580 \, a b^{7} x^{7 \, n} + 25740 \, a^{2} b^{6} x^{6 \, n} + 45045 \, a^{3} b^{5} x^{5 \, n} + 50050 \, a^{4} b^{4} x^{4 \, n} + 36036 \, a^{5} b^{3} x^{3 \, n} + 16380 \, a^{6} b^{2} x^{2 \, n} + 4290 \, a^{7} b x^{n} + 495 \, a^{8}}{6435 \, n x^{13 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-13*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

-1/6435*(1287*b^8*x^(8*n) + 8580*a*b^7*x^(7*n) + 25740*a^2*b^6*x^(6*n) + 45045*a^3*b^5*x^(5*n) + 50050*a^4*b^4
*x^(4*n) + 36036*a^5*b^3*x^(3*n) + 16380*a^6*b^2*x^(2*n) + 4290*a^7*b*x^n + 495*a^8)/(n*x^(13*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**(-1-13*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.27719, size = 153, normalized size = 1.17 \begin{align*} -\frac{1287 \, b^{8} x^{8 \, n} + 8580 \, a b^{7} x^{7 \, n} + 25740 \, a^{2} b^{6} x^{6 \, n} + 45045 \, a^{3} b^{5} x^{5 \, n} + 50050 \, a^{4} b^{4} x^{4 \, n} + 36036 \, a^{5} b^{3} x^{3 \, n} + 16380 \, a^{6} b^{2} x^{2 \, n} + 4290 \, a^{7} b x^{n} + 495 \, a^{8}}{6435 \, n x^{13 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-13*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

-1/6435*(1287*b^8*x^(8*n) + 8580*a*b^7*x^(7*n) + 25740*a^2*b^6*x^(6*n) + 45045*a^3*b^5*x^(5*n) + 50050*a^4*b^4
*x^(4*n) + 36036*a^5*b^3*x^(3*n) + 16380*a^6*b^2*x^(2*n) + 4290*a^7*b*x^n + 495*a^8)/(n*x^(13*n))