3.2574 \(\int x^{-1+n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=19 \[ \frac{\left (a+b x^n\right )^9}{9 b n} \]

[Out]

(a + b*x^n)^9/(9*b*n)

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Rubi [A]  time = 0.0045803, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{\left (a+b x^n\right )^9}{9 b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*(a + b*x^n)^8,x]

[Out]

(a + b*x^n)^9/(9*b*n)

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.004583, size = 19, normalized size = 1. \[ \frac{\left (a+b x^n\right )^9}{9 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*(a + b*x^n)^8,x]

[Out]

(a + b*x^n)^9/(9*b*n)

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Maple [B]  time = 0.023, size = 132, normalized size = 7. \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{{b}^{7}a \left ({x}^{n} \right ) ^{8}}{n}}+4\,{\frac{{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{28\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{6}}{3\,n}}+14\,{\frac{{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{5}}{n}}+14\,{\frac{{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{28\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{3}}{3\,n}}+4\,{\frac{b{a}^{7} \left ({x}^{n} \right ) ^{2}}{n}}+{\frac{{a}^{8}{x}^{n}}{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.964729, size = 23, normalized size = 1.21 \begin{align*} \frac{{\left (b x^{n} + a\right )}^{9}}{9 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(b*x^n + a)^9/(b*n)

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Fricas [B]  time = 1.3705, size = 239, normalized size = 12.58 \begin{align*} \frac{b^{8} x^{9 \, n} + 9 \, a b^{7} x^{8 \, n} + 36 \, a^{2} b^{6} x^{7 \, n} + 84 \, a^{3} b^{5} x^{6 \, n} + 126 \, a^{4} b^{4} x^{5 \, n} + 126 \, a^{5} b^{3} x^{4 \, n} + 84 \, a^{6} b^{2} x^{3 \, n} + 36 \, a^{7} b x^{2 \, n} + 9 \, a^{8} x^{n}}{9 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 86.3691, size = 133, normalized size = 7. \begin{align*} \begin{cases} \frac{a^{8} x^{n}}{n} + \frac{4 a^{7} b x^{2 n}}{n} + \frac{28 a^{6} b^{2} x^{3 n}}{3 n} + \frac{14 a^{5} b^{3} x^{4 n}}{n} + \frac{14 a^{4} b^{4} x^{5 n}}{n} + \frac{28 a^{3} b^{5} x^{6 n}}{3 n} + \frac{4 a^{2} b^{6} x^{7 n}}{n} + \frac{a b^{7} x^{8 n}}{n} + \frac{b^{8} x^{9 n}}{9 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{8} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**8*x**n/n + 4*a**7*b*x**(2*n)/n + 28*a**6*b**2*x**(3*n)/(3*n) + 14*a**5*b**3*x**(4*n)/n + 14*a**4
*b**4*x**(5*n)/n + 28*a**3*b**5*x**(6*n)/(3*n) + 4*a**2*b**6*x**(7*n)/n + a*b**7*x**(8*n)/n + b**8*x**(9*n)/(9
*n), Ne(n, 0)), ((a + b)**8*log(x), True))

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Giac [B]  time = 1.21871, size = 149, normalized size = 7.84 \begin{align*} \frac{b^{8} x^{9 \, n} + 9 \, a b^{7} x^{8 \, n} + 36 \, a^{2} b^{6} x^{7 \, n} + 84 \, a^{3} b^{5} x^{6 \, n} + 126 \, a^{4} b^{4} x^{5 \, n} + 126 \, a^{5} b^{3} x^{4 \, n} + 84 \, a^{6} b^{2} x^{3 \, n} + 36 \, a^{7} b x^{2 \, n} + 9 \, a^{8} x^{n}}{9 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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