3.2580 \(\int x^{-1-5 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=133 \[ -\frac{28 a^6 b^2 x^{-3 n}}{3 n}-\frac{28 a^5 b^3 x^{-2 n}}{n}-\frac{70 a^4 b^4 x^{-n}}{n}+\frac{28 a^2 b^6 x^n}{n}+56 a^3 b^5 \log (x)-\frac{2 a^7 b x^{-4 n}}{n}-\frac{a^8 x^{-5 n}}{5 n}+\frac{4 a b^7 x^{2 n}}{n}+\frac{b^8 x^{3 n}}{3 n} \]

[Out]

-a^8/(5*n*x^(5*n)) - (2*a^7*b)/(n*x^(4*n)) - (28*a^6*b^2)/(3*n*x^(3*n)) - (28*a^5*b^3)/(n*x^(2*n)) - (70*a^4*b
^4)/(n*x^n) + (28*a^2*b^6*x^n)/n + (4*a*b^7*x^(2*n))/n + (b^8*x^(3*n))/(3*n) + 56*a^3*b^5*Log[x]

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Rubi [A]  time = 0.0583786, antiderivative size = 133, 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} \[ -\frac{28 a^6 b^2 x^{-3 n}}{3 n}-\frac{28 a^5 b^3 x^{-2 n}}{n}-\frac{70 a^4 b^4 x^{-n}}{n}+\frac{28 a^2 b^6 x^n}{n}+56 a^3 b^5 \log (x)-\frac{2 a^7 b x^{-4 n}}{n}-\frac{a^8 x^{-5 n}}{5 n}+\frac{4 a b^7 x^{2 n}}{n}+\frac{b^8 x^{3 n}}{3 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 5*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(5*n*x^(5*n)) - (2*a^7*b)/(n*x^(4*n)) - (28*a^6*b^2)/(3*n*x^(3*n)) - (28*a^5*b^3)/(n*x^(2*n)) - (70*a^4*b
^4)/(n*x^n) + (28*a^2*b^6*x^n)/n + (4*a*b^7*x^(2*n))/n + (b^8*x^(3*n))/(3*n) + 56*a^3*b^5*Log[x]

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

Mathematica [A]  time = 0.0619468, size = 114, normalized size = 0.86 \[ \frac{-\frac{28}{3} a^6 b^2 x^{-3 n}-28 a^5 b^3 x^{-2 n}-70 a^4 b^4 x^{-n}+28 a^2 b^6 x^n+56 a^3 b^5 n \log (x)-2 a^7 b x^{-4 n}-\frac{1}{5} a^8 x^{-5 n}+4 a b^7 x^{2 n}+\frac{1}{3} b^8 x^{3 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 5*n)*(a + b*x^n)^8,x]

[Out]

(-a^8/(5*x^(5*n)) - (2*a^7*b)/x^(4*n) - (28*a^6*b^2)/(3*x^(3*n)) - (28*a^5*b^3)/x^(2*n) - (70*a^4*b^4)/x^n + 2
8*a^2*b^6*x^n + 4*a*b^7*x^(2*n) + (b^8*x^(3*n))/3 + 56*a^3*b^5*n*Log[x])/n

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Maple [A]  time = 0.023, size = 128, normalized size = 1. \begin{align*} 56\,{a}^{3}{b}^{5}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{3}}{3\,n}}+4\,{\frac{{b}^{7}a \left ({x}^{n} \right ) ^{2}}{n}}+28\,{\frac{{x}^{n}{a}^{2}{b}^{6}}{n}}-70\,{\frac{{a}^{4}{b}^{4}}{n{x}^{n}}}-28\,{\frac{{a}^{5}{b}^{3}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{28\,{a}^{6}{b}^{2}}{3\,n \left ({x}^{n} \right ) ^{3}}}-2\,{\frac{b{a}^{7}}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{{a}^{8}}{5\,n \left ({x}^{n} \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-5*n)*(a+b*x^n)^8,x)

[Out]

56*a^3*b^5*ln(x)+1/3*b^8/n*(x^n)^3+4*a*b^7/n*(x^n)^2+28*a^2*b^6*x^n/n-70*a^4*b^4/n/(x^n)-28*a^5*b^3/n/(x^n)^2-
28/3*a^6*b^2/n/(x^n)^3-2*a^7*b/n/(x^n)^4-1/5*a^8/n/(x^n)^5

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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^(-1-5*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.3377, size = 265, normalized size = 1.99 \begin{align*} \frac{840 \, a^{3} b^{5} n x^{5 \, n} \log \left (x\right ) + 5 \, b^{8} x^{8 \, n} + 60 \, a b^{7} x^{7 \, n} + 420 \, a^{2} b^{6} x^{6 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{5} b^{3} x^{3 \, n} - 140 \, a^{6} b^{2} x^{2 \, n} - 30 \, a^{7} b x^{n} - 3 \, a^{8}}{15 \, n x^{5 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-5*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/15*(840*a^3*b^5*n*x^(5*n)*log(x) + 5*b^8*x^(8*n) + 60*a*b^7*x^(7*n) + 420*a^2*b^6*x^(6*n) - 1050*a^4*b^4*x^(
4*n) - 420*a^5*b^3*x^(3*n) - 140*a^6*b^2*x^(2*n) - 30*a^7*b*x^n - 3*a^8)/(n*x^(5*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**(-1-5*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.20692, size = 157, normalized size = 1.18 \begin{align*} \frac{840 \, a^{3} b^{5} n x^{5 \, n} \log \left (x\right ) + 5 \, b^{8} x^{8 \, n} + 60 \, a b^{7} x^{7 \, n} + 420 \, a^{2} b^{6} x^{6 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{5} b^{3} x^{3 \, n} - 140 \, a^{6} b^{2} x^{2 \, n} - 30 \, a^{7} b x^{n} - 3 \, a^{8}}{15 \, n x^{5 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-5*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

1/15*(840*a^3*b^5*n*x^(5*n)*log(x) + 5*b^8*x^(8*n) + 60*a*b^7*x^(7*n) + 420*a^2*b^6*x^(6*n) - 1050*a^4*b^4*x^(
4*n) - 420*a^5*b^3*x^(3*n) - 140*a^6*b^2*x^(2*n) - 30*a^7*b*x^n - 3*a^8)/(n*x^(5*n))