3.2590 \(\int x^{-1-15 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=151 \[ -\frac {a^8 x^{-15 n}}{15 n}-\frac {4 a^7 b x^{-14 n}}{7 n}-\frac {28 a^6 b^2 x^{-13 n}}{13 n}-\frac {14 a^5 b^3 x^{-12 n}}{3 n}-\frac {70 a^4 b^4 x^{-11 n}}{11 n}-\frac {28 a^3 b^5 x^{-10 n}}{5 n}-\frac {28 a^2 b^6 x^{-9 n}}{9 n}-\frac {a b^7 x^{-8 n}}{n}-\frac {b^8 x^{-7 n}}{7 n} \]

[Out]

-1/15*a^8/n/(x^(15*n))-4/7*a^7*b/n/(x^(14*n))-28/13*a^6*b^2/n/(x^(13*n))-14/3*a^5*b^3/n/(x^(12*n))-70/11*a^4*b
^4/n/(x^(11*n))-28/5*a^3*b^5/n/(x^(10*n))-28/9*a^2*b^6/n/(x^(9*n))-a*b^7/n/(x^(8*n))-1/7*b^8/n/(x^(7*n))

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Rubi [A]  time = 0.06, antiderivative size = 151, 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} \[ -\frac {28 a^6 b^2 x^{-13 n}}{13 n}-\frac {14 a^5 b^3 x^{-12 n}}{3 n}-\frac {70 a^4 b^4 x^{-11 n}}{11 n}-\frac {28 a^3 b^5 x^{-10 n}}{5 n}-\frac {28 a^2 b^6 x^{-9 n}}{9 n}-\frac {4 a^7 b x^{-14 n}}{7 n}-\frac {a^8 x^{-15 n}}{15 n}-\frac {a b^7 x^{-8 n}}{n}-\frac {b^8 x^{-7 n}}{7 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 15*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(15*n*x^(15*n)) - (4*a^7*b)/(7*n*x^(14*n)) - (28*a^6*b^2)/(13*n*x^(13*n)) - (14*a^5*b^3)/(3*n*x^(12*n)) -
(70*a^4*b^4)/(11*n*x^(11*n)) - (28*a^3*b^5)/(5*n*x^(10*n)) - (28*a^2*b^6)/(9*n*x^(9*n)) - (a*b^7)/(n*x^(8*n))
- b^8/(7*n*x^(7*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])

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^{-1-15 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^{16}} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^8}{x^{16}}+\frac {8 a^7 b}{x^{15}}+\frac {28 a^6 b^2}{x^{14}}+\frac {56 a^5 b^3}{x^{13}}+\frac {70 a^4 b^4}{x^{12}}+\frac {56 a^3 b^5}{x^{11}}+\frac {28 a^2 b^6}{x^{10}}+\frac {8 a b^7}{x^9}+\frac {b^8}{x^8}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^8 x^{-15 n}}{15 n}-\frac {4 a^7 b x^{-14 n}}{7 n}-\frac {28 a^6 b^2 x^{-13 n}}{13 n}-\frac {14 a^5 b^3 x^{-12 n}}{3 n}-\frac {70 a^4 b^4 x^{-11 n}}{11 n}-\frac {28 a^3 b^5 x^{-10 n}}{5 n}-\frac {28 a^2 b^6 x^{-9 n}}{9 n}-\frac {a b^7 x^{-8 n}}{n}-\frac {b^8 x^{-7 n}}{7 n}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 113, normalized size = 0.75 \[ -\frac {x^{-15 n} \left (3003 a^8+25740 a^7 b x^n+97020 a^6 b^2 x^{2 n}+210210 a^5 b^3 x^{3 n}+286650 a^4 b^4 x^{4 n}+252252 a^3 b^5 x^{5 n}+140140 a^2 b^6 x^{6 n}+45045 a b^7 x^{7 n}+6435 b^8 x^{8 n}\right )}{45045 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 15*n)*(a + b*x^n)^8,x]

[Out]

-1/45045*(3003*a^8 + 25740*a^7*b*x^n + 97020*a^6*b^2*x^(2*n) + 210210*a^5*b^3*x^(3*n) + 286650*a^4*b^4*x^(4*n)
+ 252252*a^3*b^5*x^(5*n) + 140140*a^2*b^6*x^(6*n) + 45045*a*b^7*x^(7*n) + 6435*b^8*x^(8*n))/(n*x^(15*n))

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fricas [A]  time = 0.54, size = 113, normalized size = 0.75 \[ -\frac {6435 \, b^{8} x^{8 \, n} + 45045 \, a b^{7} x^{7 \, n} + 140140 \, a^{2} b^{6} x^{6 \, n} + 252252 \, a^{3} b^{5} x^{5 \, n} + 286650 \, a^{4} b^{4} x^{4 \, n} + 210210 \, a^{5} b^{3} x^{3 \, n} + 97020 \, a^{6} b^{2} x^{2 \, n} + 25740 \, a^{7} b x^{n} + 3003 \, a^{8}}{45045 \, n x^{15 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-15*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

-1/45045*(6435*b^8*x^(8*n) + 45045*a*b^7*x^(7*n) + 140140*a^2*b^6*x^(6*n) + 252252*a^3*b^5*x^(5*n) + 286650*a^
4*b^4*x^(4*n) + 210210*a^5*b^3*x^(3*n) + 97020*a^6*b^2*x^(2*n) + 25740*a^7*b*x^n + 3003*a^8)/(n*x^(15*n))

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giac [A]  time = 0.29, size = 113, normalized size = 0.75 \[ -\frac {6435 \, b^{8} x^{8 \, n} + 45045 \, a b^{7} x^{7 \, n} + 140140 \, a^{2} b^{6} x^{6 \, n} + 252252 \, a^{3} b^{5} x^{5 \, n} + 286650 \, a^{4} b^{4} x^{4 \, n} + 210210 \, a^{5} b^{3} x^{3 \, n} + 97020 \, a^{6} b^{2} x^{2 \, n} + 25740 \, a^{7} b x^{n} + 3003 \, a^{8}}{45045 \, n x^{15 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-15*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

-1/45045*(6435*b^8*x^(8*n) + 45045*a*b^7*x^(7*n) + 140140*a^2*b^6*x^(6*n) + 252252*a^3*b^5*x^(5*n) + 286650*a^
4*b^4*x^(4*n) + 210210*a^5*b^3*x^(3*n) + 97020*a^6*b^2*x^(2*n) + 25740*a^7*b*x^n + 3003*a^8)/(n*x^(15*n))

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maple [A]  time = 0.03, size = 136, normalized size = 0.90 \[ -\frac {a^{8} x^{-15 n}}{15 n}-\frac {4 a^{7} b \,x^{-14 n}}{7 n}-\frac {28 a^{6} b^{2} x^{-13 n}}{13 n}-\frac {14 a^{5} b^{3} x^{-12 n}}{3 n}-\frac {70 a^{4} b^{4} x^{-11 n}}{11 n}-\frac {28 a^{3} b^{5} x^{-10 n}}{5 n}-\frac {28 a^{2} b^{6} x^{-9 n}}{9 n}-\frac {a \,b^{7} x^{-8 n}}{n}-\frac {b^{8} x^{-7 n}}{7 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-15*n)*(b*x^n+a)^8,x)

[Out]

-1/7*b^8/n/(x^n)^7-a*b^7/n/(x^n)^8-28/9*a^2*b^6/n/(x^n)^9-28/5*a^3*b^5/n/(x^n)^10-70/11*a^4*b^4/n/(x^n)^11-14/
3*a^5*b^3/n/(x^n)^12-28/13*a^6*b^2/n/(x^n)^13-4/7*a^7*b/n/(x^n)^14-1/15*a^8/n/(x^n)^15

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maxima [A]  time = 0.54, size = 153, normalized size = 1.01 \[ -\frac {a^{8}}{15 \, n x^{15 \, n}} - \frac {4 \, a^{7} b}{7 \, n x^{14 \, n}} - \frac {28 \, a^{6} b^{2}}{13 \, n x^{13 \, n}} - \frac {14 \, a^{5} b^{3}}{3 \, n x^{12 \, n}} - \frac {70 \, a^{4} b^{4}}{11 \, n x^{11 \, n}} - \frac {28 \, a^{3} b^{5}}{5 \, n x^{10 \, n}} - \frac {28 \, a^{2} b^{6}}{9 \, n x^{9 \, n}} - \frac {a b^{7}}{n x^{8 \, n}} - \frac {b^{8}}{7 \, n x^{7 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-15*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

-1/15*a^8/(n*x^(15*n)) - 4/7*a^7*b/(n*x^(14*n)) - 28/13*a^6*b^2/(n*x^(13*n)) - 14/3*a^5*b^3/(n*x^(12*n)) - 70/
11*a^4*b^4/(n*x^(11*n)) - 28/5*a^3*b^5/(n*x^(10*n)) - 28/9*a^2*b^6/(n*x^(9*n)) - a*b^7/(n*x^(8*n)) - 1/7*b^8/(
n*x^(7*n))

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mupad [B]  time = 1.43, size = 153, normalized size = 1.01 \[ -\frac {a^8}{15\,n\,x^{15\,n}}-\frac {b^8}{7\,n\,x^{7\,n}}-\frac {28\,a^2\,b^6}{9\,n\,x^{9\,n}}-\frac {28\,a^3\,b^5}{5\,n\,x^{10\,n}}-\frac {70\,a^4\,b^4}{11\,n\,x^{11\,n}}-\frac {14\,a^5\,b^3}{3\,n\,x^{12\,n}}-\frac {28\,a^6\,b^2}{13\,n\,x^{13\,n}}-\frac {a\,b^7}{n\,x^{8\,n}}-\frac {4\,a^7\,b}{7\,n\,x^{14\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^8/x^(15*n + 1),x)

[Out]

- a^8/(15*n*x^(15*n)) - b^8/(7*n*x^(7*n)) - (28*a^2*b^6)/(9*n*x^(9*n)) - (28*a^3*b^5)/(5*n*x^(10*n)) - (70*a^4
*b^4)/(11*n*x^(11*n)) - (14*a^5*b^3)/(3*n*x^(12*n)) - (28*a^6*b^2)/(13*n*x^(13*n)) - (a*b^7)/(n*x^(8*n)) - (4*
a^7*b)/(7*n*x^(14*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**(-1-15*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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