3.2725 \(\int x^{-1-9 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=24 \[ -\frac{x^{-9 n} \left (a+b x^n\right )^9}{9 a n} \]

[Out]

-(a + b*x^n)^9/(9*a*n*x^(9*n))

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Rubi [A]  time = 0.005459, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ -\frac{x^{-9 n} \left (a+b x^n\right )^9}{9 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 9*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(9*a*n*x^(9*n))

Rule 264

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

Rubi steps

\begin{align*} \int x^{-1-9 n} \left (a+b x^n\right )^8 \, dx &=-\frac{x^{-9 n} \left (a+b x^n\right )^9}{9 a n}\\ \end{align*}

Mathematica [A]  time = 0.0097922, size = 24, normalized size = 1. \[ -\frac{x^{-9 n} \left (a+b x^n\right )^9}{9 a n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 9*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(9*a*n*x^(9*n))

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Maple [B]  time = 0., size = 136, normalized size = 5.7 \begin{align*} -{\frac{{b}^{8}}{n{x}^{n}}}-4\,{\frac{{b}^{7}a}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{28\,{a}^{2}{b}^{6}}{3\,n \left ({x}^{n} \right ) ^{3}}}-14\,{\frac{{a}^{3}{b}^{5}}{n \left ({x}^{n} \right ) ^{4}}}-14\,{\frac{{a}^{4}{b}^{4}}{n \left ({x}^{n} \right ) ^{5}}}-{\frac{28\,{a}^{5}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{6}}}-4\,{\frac{{a}^{6}{b}^{2}}{n \left ({x}^{n} \right ) ^{7}}}-{\frac{b{a}^{7}}{n \left ({x}^{n} \right ) ^{8}}}-{\frac{{a}^{8}}{9\,n \left ({x}^{n} \right ) ^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-9*n)*(a+b*x^n)^8,x)

[Out]

-b^8/n/(x^n)-4*a*b^7/n/(x^n)^2-28/3*a^2*b^6/n/(x^n)^3-14*a^3*b^5/n/(x^n)^4-14*a^4*b^4/n/(x^n)^5-28/3*a^5*b^3/n
/(x^n)^6-4*a^6*b^2/n/(x^n)^7-a^7*b/n/(x^n)^8-1/9*a^8/n/(x^n)^9

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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^(-1-9*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.40938, size = 243, normalized size = 10.12 \begin{align*} -\frac{9 \, b^{8} x^{8 \, n} + 36 \, a b^{7} x^{7 \, n} + 84 \, a^{2} b^{6} x^{6 \, n} + 126 \, a^{3} b^{5} x^{5 \, n} + 126 \, a^{4} b^{4} x^{4 \, n} + 84 \, a^{5} b^{3} x^{3 \, n} + 36 \, a^{6} b^{2} x^{2 \, n} + 9 \, a^{7} b x^{n} + a^{8}}{9 \, n x^{9 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-9*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

-1/9*(9*b^8*x^(8*n) + 36*a*b^7*x^(7*n) + 84*a^2*b^6*x^(6*n) + 126*a^3*b^5*x^(5*n) + 126*a^4*b^4*x^(4*n) + 84*a
^5*b^3*x^(3*n) + 36*a^6*b^2*x^(2*n) + 9*a^7*b*x^n + a^8)/(n*x^(9*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**(-1-9*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.18481, size = 150, normalized size = 6.25 \begin{align*} -\frac{9 \, b^{8} x^{8 \, n} + 36 \, a b^{7} x^{7 \, n} + 84 \, a^{2} b^{6} x^{6 \, n} + 126 \, a^{3} b^{5} x^{5 \, n} + 126 \, a^{4} b^{4} x^{4 \, n} + 84 \, a^{5} b^{3} x^{3 \, n} + 36 \, a^{6} b^{2} x^{2 \, n} + 9 \, a^{7} b x^{n} + a^{8}}{9 \, n x^{9 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-9*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

-1/9*(9*b^8*x^(8*n) + 36*a*b^7*x^(7*n) + 84*a^2*b^6*x^(6*n) + 126*a^3*b^5*x^(5*n) + 126*a^4*b^4*x^(4*n) + 84*a
^5*b^3*x^(3*n) + 36*a^6*b^2*x^(2*n) + 9*a^7*b*x^n + a^8)/(n*x^(9*n))