3.2583 \(\int x^{-1-8 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=140 \[ -\frac{14 a^6 b^2 x^{-6 n}}{3 n}-\frac{56 a^5 b^3 x^{-5 n}}{5 n}-\frac{35 a^4 b^4 x^{-4 n}}{2 n}-\frac{56 a^3 b^5 x^{-3 n}}{3 n}-\frac{14 a^2 b^6 x^{-2 n}}{n}-\frac{8 a^7 b x^{-7 n}}{7 n}-\frac{a^8 x^{-8 n}}{8 n}-\frac{8 a b^7 x^{-n}}{n}+b^8 \log (x) \]

[Out]

-a^8/(8*n*x^(8*n)) - (8*a^7*b)/(7*n*x^(7*n)) - (14*a^6*b^2)/(3*n*x^(6*n)) - (56*a^5*b^3)/(5*n*x^(5*n)) - (35*a
^4*b^4)/(2*n*x^(4*n)) - (56*a^3*b^5)/(3*n*x^(3*n)) - (14*a^2*b^6)/(n*x^(2*n)) - (8*a*b^7)/(n*x^n) + b^8*Log[x]

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Rubi [A]  time = 0.0588352, antiderivative size = 140, 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} \[ -\frac{14 a^6 b^2 x^{-6 n}}{3 n}-\frac{56 a^5 b^3 x^{-5 n}}{5 n}-\frac{35 a^4 b^4 x^{-4 n}}{2 n}-\frac{56 a^3 b^5 x^{-3 n}}{3 n}-\frac{14 a^2 b^6 x^{-2 n}}{n}-\frac{8 a^7 b x^{-7 n}}{7 n}-\frac{a^8 x^{-8 n}}{8 n}-\frac{8 a b^7 x^{-n}}{n}+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 8*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(8*n*x^(8*n)) - (8*a^7*b)/(7*n*x^(7*n)) - (14*a^6*b^2)/(3*n*x^(6*n)) - (56*a^5*b^3)/(5*n*x^(5*n)) - (35*a
^4*b^4)/(2*n*x^(4*n)) - (56*a^3*b^5)/(3*n*x^(3*n)) - (14*a^2*b^6)/(n*x^(2*n)) - (8*a*b^7)/(n*x^n) + b^8*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^{-1-8 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^9} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^8}{x^9}+\frac{8 a^7 b}{x^8}+\frac{28 a^6 b^2}{x^7}+\frac{56 a^5 b^3}{x^6}+\frac{70 a^4 b^4}{x^5}+\frac{56 a^3 b^5}{x^4}+\frac{28 a^2 b^6}{x^3}+\frac{8 a b^7}{x^2}+\frac{b^8}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-8 n}}{8 n}-\frac{8 a^7 b x^{-7 n}}{7 n}-\frac{14 a^6 b^2 x^{-6 n}}{3 n}-\frac{56 a^5 b^3 x^{-5 n}}{5 n}-\frac{35 a^4 b^4 x^{-4 n}}{2 n}-\frac{56 a^3 b^5 x^{-3 n}}{3 n}-\frac{14 a^2 b^6 x^{-2 n}}{n}-\frac{8 a b^7 x^{-n}}{n}+b^8 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0902527, size = 108, normalized size = 0.77 \[ b^8 \log (x)-\frac{a x^{-8 n} \left (3920 a^5 b^2 x^{2 n}+9408 a^4 b^3 x^{3 n}+14700 a^3 b^4 x^{4 n}+15680 a^2 b^5 x^{5 n}+960 a^6 b x^n+105 a^7+11760 a b^6 x^{6 n}+6720 b^7 x^{7 n}\right )}{840 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(105*a^7 + 960*a^6*b*x^n + 3920*a^5*b^2*x^(2*n) + 9408*a^4*b^3*x^(3*n) + 14700*a^3*b^4*x^(4*n) + 15680*a^2
*b^5*x^(5*n) + 11760*a*b^6*x^(6*n) + 6720*b^7*x^(7*n)))/(840*n*x^(8*n)) + b^8*Log[x]

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Maple [A]  time = 0.023, size = 129, normalized size = 0.9 \begin{align*}{b}^{8}\ln \left ( x \right ) -8\,{\frac{{b}^{7}a}{n{x}^{n}}}-14\,{\frac{{a}^{2}{b}^{6}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{56\,{a}^{3}{b}^{5}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{35\,{a}^{4}{b}^{4}}{2\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{56\,{a}^{5}{b}^{3}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{14\,{a}^{6}{b}^{2}}{3\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{8\,b{a}^{7}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{{a}^{8}}{8\,n \left ({x}^{n} \right ) ^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^8*ln(x)-8*a*b^7/n/(x^n)-14*a^2*b^6/n/(x^n)^2-56/3*a^3*b^5/n/(x^n)^3-35/2*a^4*b^4/n/(x^n)^4-56/5*a^5*b^3/n/(x
^n)^5-14/3*a^6*b^2/n/(x^n)^6-8/7*a^7*b/n/(x^n)^7-1/8*a^8/n/(x^n)^8

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.04961, size = 285, normalized size = 2.04 \begin{align*} \frac{840 \, b^{8} n x^{8 \, n} \log \left (x\right ) - 6720 \, a b^{7} x^{7 \, n} - 11760 \, a^{2} b^{6} x^{6 \, n} - 15680 \, a^{3} b^{5} x^{5 \, n} - 14700 \, a^{4} b^{4} x^{4 \, n} - 9408 \, a^{5} b^{3} x^{3 \, n} - 3920 \, a^{6} b^{2} x^{2 \, n} - 960 \, a^{7} b x^{n} - 105 \, a^{8}}{840 \, n x^{8 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(840*b^8*n*x^(8*n)*log(x) - 6720*a*b^7*x^(7*n) - 11760*a^2*b^6*x^(6*n) - 15680*a^3*b^5*x^(5*n) - 14700*a
^4*b^4*x^(4*n) - 9408*a^5*b^3*x^(3*n) - 3920*a^6*b^2*x^(2*n) - 960*a^7*b*x^n - 105*a^8)/(n*x^(8*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**(-1-8*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.21854, size = 157, normalized size = 1.12 \begin{align*} \frac{840 \, b^{8} n x^{8 \, n} \log \left (x\right ) - 6720 \, a b^{7} x^{7 \, n} - 11760 \, a^{2} b^{6} x^{6 \, n} - 15680 \, a^{3} b^{5} x^{5 \, n} - 14700 \, a^{4} b^{4} x^{4 \, n} - 9408 \, a^{5} b^{3} x^{3 \, n} - 3920 \, a^{6} b^{2} x^{2 \, n} - 960 \, a^{7} b x^{n} - 105 \, a^{8}}{840 \, n x^{8 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*(840*b^8*n*x^(8*n)*log(x) - 6720*a*b^7*x^(7*n) - 11760*a^2*b^6*x^(6*n) - 15680*a^3*b^5*x^(5*n) - 14700*a
^4*b^4*x^(4*n) - 9408*a^5*b^3*x^(3*n) - 3920*a^6*b^2*x^(2*n) - 960*a^7*b*x^n - 105*a^8)/(n*x^(8*n))