3.28.25 \(\int x^{-1-9 n} (a+b x^n)^8 \, dx\) [2725]

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

[Out]

-1/9*(a+b*x^n)^9/a/n/(x^(9*n))

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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {270} \begin {gather*} -\frac {x^{-9 n} \left (a+b x^n\right )^9}{9 a n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 9*n)*(a + b*x^n)^8,x]

[Out]

-1/9*(a + b*x^n)^9/(a*n*x^(9*n))

Rule 270

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(24)=48\).
time = 0.02, size = 113, normalized size = 4.71 \begin {gather*} \frac {x^{-9 n} \left (-a^8-9 a^7 b x^n-36 a^6 b^2 x^{2 n}-84 a^5 b^3 x^{3 n}-126 a^4 b^4 x^{4 n}-126 a^3 b^5 x^{5 n}-84 a^2 b^6 x^{6 n}-36 a b^7 x^{7 n}-9 b^8 x^{8 n}\right )}{9 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(24)=48\).
time = 0.20, size = 136, normalized size = 5.67

method	result	size
risch	\(-\frac {b^{8} x^{-n}}{n}-\frac {4 a \,b^{7} x^{-2 n}}{n}-\frac {28 a^{2} b^{6} x^{-3 n}}{3 n}-\frac {14 a^{3} b^{5} x^{-4 n}}{n}-\frac {14 a^{4} b^{4} x^{-5 n}}{n}-\frac {28 a^{5} b^{3} x^{-6 n}}{3 n}-\frac {4 a^{6} b^{2} x^{-7 n}}{n}-\frac {a^{7} b \,x^{-8 n}}{n}-\frac {a^{8} x^{-9 n}}{9 n}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 [B] Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (24) = 48\).
time = 0.29, size = 151, normalized size = 6.29 \begin {gather*} -\frac {a^{8}}{9 \, n x^{9 \, n}} - \frac {a^{7} b}{n x^{8 \, n}} - \frac {4 \, a^{6} b^{2}}{n x^{7 \, n}} - \frac {28 \, a^{5} b^{3}}{3 \, n x^{6 \, n}} - \frac {14 \, a^{4} b^{4}}{n x^{5 \, n}} - \frac {14 \, a^{3} b^{5}}{n x^{4 \, n}} - \frac {28 \, a^{2} b^{6}}{3 \, n x^{3 \, n}} - \frac {4 \, a b^{7}}{n x^{2 \, n}} - \frac {b^{8}}{n x^{n}} \end {gather*}

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]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (24) = 48\).
time = 0.38, size = 111, normalized size = 4.62 \begin {gather*} -\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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (19) = 38\).
time = 36.95, size = 134, normalized size = 5.58 \begin {gather*} \begin {cases} - \frac {a^{8} x^{- 9 n}}{9 n} - \frac {a^{7} b x^{- 8 n}}{n} - \frac {4 a^{6} b^{2} x^{- 7 n}}{n} - \frac {28 a^{5} b^{3} x^{- 6 n}}{3 n} - \frac {14 a^{4} b^{4} x^{- 5 n}}{n} - \frac {14 a^{3} b^{5} x^{- 4 n}}{n} - \frac {28 a^{2} b^{6} x^{- 3 n}}{3 n} - \frac {4 a b^{7} x^{- 2 n}}{n} - \frac {b^{8} x^{- n}}{n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (24) = 48\).
time = 0.62, size = 111, normalized size = 4.62 \begin {gather*} -\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 {gather*}

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))

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Mupad [B]
time = 0.00, size = 151, normalized size = 6.29 \begin {gather*} -\frac {a^8}{9\,n\,x^{9\,n}}-\frac {b^8}{n\,x^n}-\frac {28\,a^2\,b^6}{3\,n\,x^{3\,n}}-\frac {14\,a^3\,b^5}{n\,x^{4\,n}}-\frac {14\,a^4\,b^4}{n\,x^{5\,n}}-\frac {28\,a^5\,b^3}{3\,n\,x^{6\,n}}-\frac {4\,a^6\,b^2}{n\,x^{7\,n}}-\frac {4\,a\,b^7}{n\,x^{2\,n}}-\frac {a^7\,b}{n\,x^{8\,n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^8/x^(9*n + 1),x)

[Out]

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

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