3.2576 \(\int x^{-1-n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=135 \[ -\frac {a^8 x^{-n}}{n}+8 a^7 b \log (x)+\frac {28 a^6 b^2 x^n}{n}+\frac {28 a^5 b^3 x^{2 n}}{n}+\frac {70 a^4 b^4 x^{3 n}}{3 n}+\frac {14 a^3 b^5 x^{4 n}}{n}+\frac {28 a^2 b^6 x^{5 n}}{5 n}+\frac {4 a b^7 x^{6 n}}{3 n}+\frac {b^8 x^{7 n}}{7 n} \]

[Out]

-a^8/n/(x^n)+28*a^6*b^2*x^n/n+28*a^5*b^3*x^(2*n)/n+70/3*a^4*b^4*x^(3*n)/n+14*a^3*b^5*x^(4*n)/n+28/5*a^2*b^6*x^
(5*n)/n+4/3*a*b^7*x^(6*n)/n+1/7*b^8*x^(7*n)/n+8*a^7*b*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} \[ \frac {28 a^6 b^2 x^n}{n}+\frac {28 a^5 b^3 x^{2 n}}{n}+\frac {70 a^4 b^4 x^{3 n}}{3 n}+\frac {14 a^3 b^5 x^{4 n}}{n}+\frac {28 a^2 b^6 x^{5 n}}{5 n}+8 a^7 b \log (x)-\frac {a^8 x^{-n}}{n}+\frac {4 a b^7 x^{6 n}}{3 n}+\frac {b^8 x^{7 n}}{7 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*(a + b*x^n)^8,x]

[Out]

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

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Mathematica [A]  time = 0.10, size = 116, normalized size = 0.86 \[ \frac {-a^8 x^{-n}+8 a^7 b n \log (x)+28 a^6 b^2 x^n+28 a^5 b^3 x^{2 n}+\frac {70}{3} a^4 b^4 x^{3 n}+14 a^3 b^5 x^{4 n}+\frac {28}{5} a^2 b^6 x^{5 n}+\frac {4}{3} a b^7 x^{6 n}+\frac {1}{7} b^8 x^{7 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n)*(a + b*x^n)^8,x]

[Out]

(-(a^8/x^n) + 28*a^6*b^2*x^n + 28*a^5*b^3*x^(2*n) + (70*a^4*b^4*x^(3*n))/3 + 14*a^3*b^5*x^(4*n) + (28*a^2*b^6*
x^(5*n))/5 + (4*a*b^7*x^(6*n))/3 + (b^8*x^(7*n))/7 + 8*a^7*b*n*Log[x])/n

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fricas [A]  time = 0.61, size = 114, normalized size = 0.84 \[ \frac {840 \, a^{7} b n x^{n} \log \relax (x) + 15 \, b^{8} x^{8 \, n} + 140 \, a b^{7} x^{7 \, n} + 588 \, a^{2} b^{6} x^{6 \, n} + 1470 \, a^{3} b^{5} x^{5 \, n} + 2450 \, a^{4} b^{4} x^{4 \, n} + 2940 \, a^{5} b^{3} x^{3 \, n} + 2940 \, a^{6} b^{2} x^{2 \, n} - 105 \, a^{8}}{105 \, n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(840*a^7*b*n*x^n*log(x) + 15*b^8*x^(8*n) + 140*a*b^7*x^(7*n) + 588*a^2*b^6*x^(6*n) + 1470*a^3*b^5*x^(5*n
) + 2450*a^4*b^4*x^(4*n) + 2940*a^5*b^3*x^(3*n) + 2940*a^6*b^2*x^(2*n) - 105*a^8)/(n*x^n)

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giac [A]  time = 0.28, size = 114, normalized size = 0.84 \[ \frac {840 \, a^{7} b n x^{n} \log \relax (x) + 15 \, b^{8} x^{8 \, n} + 140 \, a b^{7} x^{7 \, n} + 588 \, a^{2} b^{6} x^{6 \, n} + 1470 \, a^{3} b^{5} x^{5 \, n} + 2450 \, a^{4} b^{4} x^{4 \, n} + 2940 \, a^{5} b^{3} x^{3 \, n} + 2940 \, a^{6} b^{2} x^{2 \, n} - 105 \, a^{8}}{105 \, n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(840*a^7*b*n*x^n*log(x) + 15*b^8*x^(8*n) + 140*a*b^7*x^(7*n) + 588*a^2*b^6*x^(6*n) + 1470*a^3*b^5*x^(5*n
) + 2450*a^4*b^4*x^(4*n) + 2940*a^5*b^3*x^(3*n) + 2940*a^6*b^2*x^(2*n) - 105*a^8)/(n*x^n)

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maple [A]  time = 0.03, size = 128, normalized size = 0.95 \[ 8 a^{7} b \ln \relax (x )-\frac {a^{8} x^{-n}}{n}+\frac {28 a^{6} b^{2} x^{n}}{n}+\frac {28 a^{5} b^{3} x^{2 n}}{n}+\frac {70 a^{4} b^{4} x^{3 n}}{3 n}+\frac {14 a^{3} b^{5} x^{4 n}}{n}+\frac {28 a^{2} b^{6} x^{5 n}}{5 n}+\frac {4 a \,b^{7} x^{6 n}}{3 n}+\frac {b^{8} x^{7 n}}{7 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*a^7*b*ln(x)+1/7*b^8/n*(x^n)^7+4/3*a*b^7/n*(x^n)^6+28/5*a^2*b^6/n*(x^n)^5+14*a^3*b^5/n*(x^n)^4+70/3*a^4*b^4/n
*(x^n)^3+28*a^5*b^3/n*(x^n)^2+28*a^6*b^2*x^n/n-a^8/n/(x^n)

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maxima [A]  time = 0.50, size = 127, normalized size = 0.94 \[ 8 \, a^{7} b \log \relax (x) + \frac {b^{8} x^{7 \, n}}{7 \, n} + \frac {4 \, a b^{7} x^{6 \, n}}{3 \, n} + \frac {28 \, a^{2} b^{6} x^{5 \, n}}{5 \, n} + \frac {14 \, a^{3} b^{5} x^{4 \, n}}{n} + \frac {70 \, a^{4} b^{4} x^{3 \, n}}{3 \, n} + \frac {28 \, a^{5} b^{3} x^{2 \, n}}{n} + \frac {28 \, a^{6} b^{2} x^{n}}{n} - \frac {a^{8}}{n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*a^7*b*log(x) + 1/7*b^8*x^(7*n)/n + 4/3*a*b^7*x^(6*n)/n + 28/5*a^2*b^6*x^(5*n)/n + 14*a^3*b^5*x^(4*n)/n + 70/
3*a^4*b^4*x^(3*n)/n + 28*a^5*b^3*x^(2*n)/n + 28*a^6*b^2*x^n/n - a^8/(n*x^n)

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mupad [B]  time = 1.72, size = 127, normalized size = 0.94 \[ 8\,a^7\,b\,\ln \relax (x)-\frac {a^8}{n\,x^n}+\frac {b^8\,x^{7\,n}}{7\,n}+\frac {28\,a^5\,b^3\,x^{2\,n}}{n}+\frac {70\,a^4\,b^4\,x^{3\,n}}{3\,n}+\frac {14\,a^3\,b^5\,x^{4\,n}}{n}+\frac {28\,a^2\,b^6\,x^{5\,n}}{5\,n}+\frac {4\,a\,b^7\,x^{6\,n}}{3\,n}+\frac {28\,a^6\,b^2\,x^n}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^8/x^(n + 1),x)

[Out]

8*a^7*b*log(x) - a^8/(n*x^n) + (b^8*x^(7*n))/(7*n) + (28*a^5*b^3*x^(2*n))/n + (70*a^4*b^4*x^(3*n))/(3*n) + (14
*a^3*b^5*x^(4*n))/n + (28*a^2*b^6*x^(5*n))/(5*n) + (4*a*b^7*x^(6*n))/(3*n) + (28*a^6*b^2*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-n)*(a+b*x**n)**8,x)

[Out]

Timed out

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