3.2579 \(\int x^{-1-4 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=135 \[ -\frac{14 a^6 b^2 x^{-2 n}}{n}-\frac{56 a^5 b^3 x^{-n}}{n}+\frac{56 a^3 b^5 x^n}{n}+\frac{14 a^2 b^6 x^{2 n}}{n}+70 a^4 b^4 \log (x)-\frac{8 a^7 b x^{-3 n}}{3 n}-\frac{a^8 x^{-4 n}}{4 n}+\frac{8 a b^7 x^{3 n}}{3 n}+\frac{b^8 x^{4 n}}{4 n} \]

[Out]

-a^8/(4*n*x^(4*n)) - (8*a^7*b)/(3*n*x^(3*n)) - (14*a^6*b^2)/(n*x^(2*n)) - (56*a^5*b^3)/(n*x^n) + (56*a^3*b^5*x
^n)/n + (14*a^2*b^6*x^(2*n))/n + (8*a*b^7*x^(3*n))/(3*n) + (b^8*x^(4*n))/(4*n) + 70*a^4*b^4*Log[x]

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Rubi [A]  time = 0.059085, antiderivative size = 135, 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} \[ -\frac{14 a^6 b^2 x^{-2 n}}{n}-\frac{56 a^5 b^3 x^{-n}}{n}+\frac{56 a^3 b^5 x^n}{n}+\frac{14 a^2 b^6 x^{2 n}}{n}+70 a^4 b^4 \log (x)-\frac{8 a^7 b x^{-3 n}}{3 n}-\frac{a^8 x^{-4 n}}{4 n}+\frac{8 a b^7 x^{3 n}}{3 n}+\frac{b^8 x^{4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 4*n)*(a + b*x^n)^8,x]

[Out]

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

Mathematica [A]  time = 0.0584527, size = 116, normalized size = 0.86 \[ \frac{-14 a^6 b^2 x^{-2 n}-56 a^5 b^3 x^{-n}+56 a^3 b^5 x^n+14 a^2 b^6 x^{2 n}+70 a^4 b^4 n \log (x)-\frac{8}{3} a^7 b x^{-3 n}-\frac{1}{4} a^8 x^{-4 n}+\frac{8}{3} a b^7 x^{3 n}+\frac{1}{4} b^8 x^{4 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 4*n)*(a + b*x^n)^8,x]

[Out]

(-a^8/(4*x^(4*n)) - (8*a^7*b)/(3*x^(3*n)) - (14*a^6*b^2)/x^(2*n) - (56*a^5*b^3)/x^n + 56*a^3*b^5*x^n + 14*a^2*
b^6*x^(2*n) + (8*a*b^7*x^(3*n))/3 + (b^8*x^(4*n))/4 + 70*a^4*b^4*n*Log[x])/n

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Maple [A]  time = 0.026, size = 128, normalized size = 1. \begin{align*} 70\,{a}^{4}{b}^{4}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{4}}{4\,n}}+{\frac{8\,{b}^{7}a \left ({x}^{n} \right ) ^{3}}{3\,n}}+14\,{\frac{{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{2}}{n}}+56\,{\frac{{x}^{n}{a}^{3}{b}^{5}}{n}}-56\,{\frac{{a}^{5}{b}^{3}}{n{x}^{n}}}-14\,{\frac{{a}^{6}{b}^{2}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{8\,b{a}^{7}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{{a}^{8}}{4\,n \left ({x}^{n} \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-4*n)*(a+b*x^n)^8,x)

[Out]

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

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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^(-1-4*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.38866, size = 263, normalized size = 1.95 \begin{align*} \frac{840 \, a^{4} b^{4} n x^{4 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 32 \, a b^{7} x^{7 \, n} + 168 \, a^{2} b^{6} x^{6 \, n} + 672 \, a^{3} b^{5} x^{5 \, n} - 672 \, a^{5} b^{3} x^{3 \, n} - 168 \, a^{6} b^{2} x^{2 \, n} - 32 \, a^{7} b x^{n} - 3 \, a^{8}}{12 \, n x^{4 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/12*(840*a^4*b^4*n*x^(4*n)*log(x) + 3*b^8*x^(8*n) + 32*a*b^7*x^(7*n) + 168*a^2*b^6*x^(6*n) + 672*a^3*b^5*x^(5
*n) - 672*a^5*b^3*x^(3*n) - 168*a^6*b^2*x^(2*n) - 32*a^7*b*x^n - 3*a^8)/(n*x^(4*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**(-1-4*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.22448, size = 157, normalized size = 1.16 \begin{align*} \frac{840 \, a^{4} b^{4} n x^{4 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 32 \, a b^{7} x^{7 \, n} + 168 \, a^{2} b^{6} x^{6 \, n} + 672 \, a^{3} b^{5} x^{5 \, n} - 672 \, a^{5} b^{3} x^{3 \, n} - 168 \, a^{6} b^{2} x^{2 \, n} - 32 \, a^{7} b x^{n} - 3 \, a^{8}}{12 \, n x^{4 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

1/12*(840*a^4*b^4*n*x^(4*n)*log(x) + 3*b^8*x^(8*n) + 32*a*b^7*x^(7*n) + 168*a^2*b^6*x^(6*n) + 672*a^3*b^5*x^(5
*n) - 672*a^5*b^3*x^(3*n) - 168*a^6*b^2*x^(2*n) - 32*a^7*b*x^n - 3*a^8)/(n*x^(4*n))