3.2577 \(\int x^{-1-2 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=135 \[ -\frac {a^8 x^{-2 n}}{2 n}-\frac {8 a^7 b x^{-n}}{n}+28 a^6 b^2 \log (x)+\frac {56 a^5 b^3 x^n}{n}+\frac {35 a^4 b^4 x^{2 n}}{n}+\frac {56 a^3 b^5 x^{3 n}}{3 n}+\frac {7 a^2 b^6 x^{4 n}}{n}+\frac {8 a b^7 x^{5 n}}{5 n}+\frac {b^8 x^{6 n}}{6 n} \]

[Out]

-1/2*a^8/n/(x^(2*n))-8*a^7*b/n/(x^n)+56*a^5*b^3*x^n/n+35*a^4*b^4*x^(2*n)/n+56/3*a^3*b^5*x^(3*n)/n+7*a^2*b^6*x^
(4*n)/n+8/5*a*b^7*x^(5*n)/n+1/6*b^8*x^(6*n)/n+28*a^6*b^2*ln(x)

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Rubi [A]  time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, 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} \[ \frac {56 a^5 b^3 x^n}{n}+\frac {35 a^4 b^4 x^{2 n}}{n}+\frac {56 a^3 b^5 x^{3 n}}{3 n}+\frac {7 a^2 b^6 x^{4 n}}{n}+28 a^6 b^2 \log (x)-\frac {8 a^7 b x^{-n}}{n}-\frac {a^8 x^{-2 n}}{2 n}+\frac {8 a b^7 x^{5 n}}{5 n}+\frac {b^8 x^{6 n}}{6 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(2*n*x^(2*n)) - (8*a^7*b)/(n*x^n) + (56*a^5*b^3*x^n)/n + (35*a^4*b^4*x^(2*n))/n + (56*a^3*b^5*x^(3*n))/(3
*n) + (7*a^2*b^6*x^(4*n))/n + (8*a*b^7*x^(5*n))/(5*n) + (b^8*x^(6*n))/(6*n) + 28*a^6*b^2*Log[x]

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])

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]]

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 116, normalized size = 0.86 \[ \frac {-\frac {1}{2} a^8 x^{-2 n}-8 a^7 b x^{-n}+28 a^6 b^2 n \log (x)+56 a^5 b^3 x^n+35 a^4 b^4 x^{2 n}+\frac {56}{3} a^3 b^5 x^{3 n}+7 a^2 b^6 x^{4 n}+\frac {8}{5} a b^7 x^{5 n}+\frac {1}{6} b^8 x^{6 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 2*n)*(a + b*x^n)^8,x]

[Out]

(-1/2*a^8/x^(2*n) - (8*a^7*b)/x^n + 56*a^5*b^3*x^n + 35*a^4*b^4*x^(2*n) + (56*a^3*b^5*x^(3*n))/3 + 7*a^2*b^6*x
^(4*n) + (8*a*b^7*x^(5*n))/5 + (b^8*x^(6*n))/6 + 28*a^6*b^2*n*Log[x])/n

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fricas [A]  time = 0.87, size = 116, normalized size = 0.86 \[ \frac {840 \, a^{6} b^{2} n x^{2 \, n} \log \relax (x) + 5 \, b^{8} x^{8 \, n} + 48 \, a b^{7} x^{7 \, n} + 210 \, a^{2} b^{6} x^{6 \, n} + 560 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 1680 \, a^{5} b^{3} x^{3 \, n} - 240 \, a^{7} b x^{n} - 15 \, a^{8}}{30 \, n x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/30*(840*a^6*b^2*n*x^(2*n)*log(x) + 5*b^8*x^(8*n) + 48*a*b^7*x^(7*n) + 210*a^2*b^6*x^(6*n) + 560*a^3*b^5*x^(5
*n) + 1050*a^4*b^4*x^(4*n) + 1680*a^5*b^3*x^(3*n) - 240*a^7*b*x^n - 15*a^8)/(n*x^(2*n))

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giac [A]  time = 0.29, size = 116, normalized size = 0.86 \[ \frac {840 \, a^{6} b^{2} n x^{2 \, n} \log \relax (x) + 5 \, b^{8} x^{8 \, n} + 48 \, a b^{7} x^{7 \, n} + 210 \, a^{2} b^{6} x^{6 \, n} + 560 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 1680 \, a^{5} b^{3} x^{3 \, n} - 240 \, a^{7} b x^{n} - 15 \, a^{8}}{30 \, n x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

1/30*(840*a^6*b^2*n*x^(2*n)*log(x) + 5*b^8*x^(8*n) + 48*a*b^7*x^(7*n) + 210*a^2*b^6*x^(6*n) + 560*a^3*b^5*x^(5
*n) + 1050*a^4*b^4*x^(4*n) + 1680*a^5*b^3*x^(3*n) - 240*a^7*b*x^n - 15*a^8)/(n*x^(2*n))

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maple [A]  time = 0.03, size = 128, normalized size = 0.95 \[ 28 a^{6} b^{2} \ln \relax (x )-\frac {a^{8} x^{-2 n}}{2 n}-\frac {8 a^{7} b \,x^{-n}}{n}+\frac {56 a^{5} b^{3} x^{n}}{n}+\frac {35 a^{4} b^{4} x^{2 n}}{n}+\frac {56 a^{3} b^{5} x^{3 n}}{3 n}+\frac {7 a^{2} b^{6} x^{4 n}}{n}+\frac {8 a \,b^{7} x^{5 n}}{5 n}+\frac {b^{8} x^{6 n}}{6 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-2*n)*(b*x^n+a)^8,x)

[Out]

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

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maxima [A]  time = 0.52, size = 129, normalized size = 0.96 \[ 28 \, a^{6} b^{2} \log \relax (x) + \frac {b^{8} x^{6 \, n}}{6 \, n} + \frac {8 \, a b^{7} x^{5 \, n}}{5 \, n} + \frac {7 \, a^{2} b^{6} x^{4 \, n}}{n} + \frac {56 \, a^{3} b^{5} x^{3 \, n}}{3 \, n} + \frac {35 \, a^{4} b^{4} x^{2 \, n}}{n} + \frac {56 \, a^{5} b^{3} x^{n}}{n} - \frac {a^{8}}{2 \, n x^{2 \, n}} - \frac {8 \, a^{7} b}{n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.60, size = 129, normalized size = 0.96 \[ \frac {b^8\,x^{6\,n}}{6\,n}-\frac {a^8}{2\,n\,x^{2\,n}}+28\,a^6\,b^2\,\ln \relax (x)+\frac {35\,a^4\,b^4\,x^{2\,n}}{n}+\frac {56\,a^3\,b^5\,x^{3\,n}}{3\,n}+\frac {7\,a^2\,b^6\,x^{4\,n}}{n}-\frac {8\,a^7\,b}{n\,x^n}+\frac {8\,a\,b^7\,x^{5\,n}}{5\,n}+\frac {56\,a^5\,b^3\,x^n}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^8/x^(2*n + 1),x)

[Out]

(b^8*x^(6*n))/(6*n) - a^8/(2*n*x^(2*n)) + 28*a^6*b^2*log(x) + (35*a^4*b^4*x^(2*n))/n + (56*a^3*b^5*x^(3*n))/(3
*n) + (7*a^2*b^6*x^(4*n))/n - (8*a^7*b)/(n*x^n) + (8*a*b^7*x^(5*n))/(5*n) + (56*a^5*b^3*x^n)/n

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-2*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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