3.2563 \(\int x^{-1-8 n} (a+b x^n)^5 \, dx\)

Optimal. Leaf size=77 \[ -\frac {b^2 x^{-6 n} \left (a+b x^n\right )^6}{168 a^3 n}+\frac {b x^{-7 n} \left (a+b x^n\right )^6}{28 a^2 n}-\frac {x^{-8 n} \left (a+b x^n\right )^6}{8 a n} \]

[Out]

-1/8*(a+b*x^n)^6/a/n/(x^(8*n))+1/28*b*(a+b*x^n)^6/a^2/n/(x^(7*n))-1/168*b^2*(a+b*x^n)^6/a^3/n/(x^(6*n))

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {266, 45, 37} \[ -\frac {b^2 x^{-6 n} \left (a+b x^n\right )^6}{168 a^3 n}+\frac {b x^{-7 n} \left (a+b x^n\right )^6}{28 a^2 n}-\frac {x^{-8 n} \left (a+b x^n\right )^6}{8 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 8*n)*(a + b*x^n)^5,x]

[Out]

-(a + b*x^n)^6/(8*a*n*x^(8*n)) + (b*(a + b*x^n)^6)/(28*a^2*n*x^(7*n)) - (b^2*(a + b*x^n)^6)/(168*a^3*n*x^(6*n)
)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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^{-1-8 n} \left (a+b x^n\right )^5 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^9} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-8 n} \left (a+b x^n\right )^6}{8 a n}-\frac {b \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,x^n\right )}{4 a n}\\ &=-\frac {x^{-8 n} \left (a+b x^n\right )^6}{8 a n}+\frac {b x^{-7 n} \left (a+b x^n\right )^6}{28 a^2 n}+\frac {b^2 \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^n\right )}{28 a^2 n}\\ &=-\frac {x^{-8 n} \left (a+b x^n\right )^6}{8 a n}+\frac {b x^{-7 n} \left (a+b x^n\right )^6}{28 a^2 n}-\frac {b^2 x^{-6 n} \left (a+b x^n\right )^6}{168 a^3 n}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 74, normalized size = 0.96 \[ -\frac {x^{-8 n} \left (21 a^5+120 a^4 b x^n+280 a^3 b^2 x^{2 n}+336 a^2 b^3 x^{3 n}+210 a b^4 x^{4 n}+56 b^5 x^{5 n}\right )}{168 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 8*n)*(a + b*x^n)^5,x]

[Out]

-1/168*(21*a^5 + 120*a^4*b*x^n + 280*a^3*b^2*x^(2*n) + 336*a^2*b^3*x^(3*n) + 210*a*b^4*x^(4*n) + 56*b^5*x^(5*n
))/(n*x^(8*n))

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 74, normalized size = 0.96 \[ -\frac {56 \, b^{5} x^{5 \, n} + 210 \, a b^{4} x^{4 \, n} + 336 \, a^{2} b^{3} x^{3 \, n} + 280 \, a^{3} b^{2} x^{2 \, n} + 120 \, a^{4} b x^{n} + 21 \, a^{5}}{168 \, n x^{8 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-8*n)*(a+b*x^n)^5,x, algorithm="fricas")

[Out]

-1/168*(56*b^5*x^(5*n) + 210*a*b^4*x^(4*n) + 336*a^2*b^3*x^(3*n) + 280*a^3*b^2*x^(2*n) + 120*a^4*b*x^n + 21*a^
5)/(n*x^(8*n))

________________________________________________________________________________________

giac [A]  time = 0.24, size = 74, normalized size = 0.96 \[ -\frac {56 \, b^{5} x^{5 \, n} + 210 \, a b^{4} x^{4 \, n} + 336 \, a^{2} b^{3} x^{3 \, n} + 280 \, a^{3} b^{2} x^{2 \, n} + 120 \, a^{4} b x^{n} + 21 \, a^{5}}{168 \, n x^{8 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-8*n)*(a+b*x^n)^5,x, algorithm="giac")

[Out]

-1/168*(56*b^5*x^(5*n) + 210*a*b^4*x^(4*n) + 336*a^2*b^3*x^(3*n) + 280*a^3*b^2*x^(2*n) + 120*a^4*b*x^n + 21*a^
5)/(n*x^(8*n))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 88, normalized size = 1.14 \[ -\frac {a^{5} x^{-8 n}}{8 n}-\frac {5 a^{4} b \,x^{-7 n}}{7 n}-\frac {5 a^{3} b^{2} x^{-6 n}}{3 n}-\frac {2 a^{2} b^{3} x^{-5 n}}{n}-\frac {5 a \,b^{4} x^{-4 n}}{4 n}-\frac {b^{5} x^{-3 n}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-8*n)*(b*x^n+a)^5,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.48, size = 99, normalized size = 1.29 \[ -\frac {a^{5}}{8 \, n x^{8 \, n}} - \frac {5 \, a^{4} b}{7 \, n x^{7 \, n}} - \frac {5 \, a^{3} b^{2}}{3 \, n x^{6 \, n}} - \frac {2 \, a^{2} b^{3}}{n x^{5 \, n}} - \frac {5 \, a b^{4}}{4 \, n x^{4 \, n}} - \frac {b^{5}}{3 \, n x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-8*n)*(a+b*x^n)^5,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.36, size = 99, normalized size = 1.29 \[ -\frac {a^5}{8\,n\,x^{8\,n}}-\frac {b^5}{3\,n\,x^{3\,n}}-\frac {2\,a^2\,b^3}{n\,x^{5\,n}}-\frac {5\,a^3\,b^2}{3\,n\,x^{6\,n}}-\frac {5\,a\,b^4}{4\,n\,x^{4\,n}}-\frac {5\,a^4\,b}{7\,n\,x^{7\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^5/x^(8*n + 1),x)

[Out]

- a^5/(8*n*x^(8*n)) - b^5/(3*n*x^(3*n)) - (2*a^2*b^3)/(n*x^(5*n)) - (5*a^3*b^2)/(3*n*x^(6*n)) - (5*a*b^4)/(4*n
*x^(4*n)) - (5*a^4*b)/(7*n*x^(7*n))

________________________________________________________________________________________

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**(-1-8*n)*(a+b*x**n)**5,x)

[Out]

Timed out

________________________________________________________________________________________