3.26.63 \(\int x^{-1-8 n} (a+b x^n)^5 \, dx\) [2563]

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

[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))

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Rubi [A]
time = 0.02, 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 = {272, 47, 37} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 8*n)*(a + b*x^n)^5,x]

[Out]

-1/8*(a + b*x^n)^6/(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 47

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 272

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 {\text {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 \text {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 \text {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*}

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Mathematica [A]
time = 0.04, size = 74, normalized size = 0.96 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-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))/(16
8*n*x^(8*n))

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Maple [A]
time = 0.24, size = 88, normalized size = 1.14

method	result	size
risch	\(-\frac {b^{5} x^{-3 n}}{3 n}-\frac {5 a \,b^{4} x^{-4 n}}{4 n}-\frac {2 a^{2} b^{3} x^{-5 n}}{n}-\frac {5 a^{3} b^{2} x^{-6 n}}{3 n}-\frac {5 a^{4} b \,x^{-7 n}}{7 n}-\frac {a^{5} x^{-8 n}}{8 n}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]
time = 0.29, size = 99, normalized size = 1.29 \begin {gather*} -\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}} \end {gather*}

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))

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Fricas [A]
time = 0.40, size = 74, normalized size = 0.96 \begin {gather*} -\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}} \end {gather*}

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))

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Sympy [A]
time = 6.84, size = 95, normalized size = 1.23 \begin {gather*} \begin {cases} - \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} & \text {for}\: n \neq 0 \\\left (a + b\right )^{5} \log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**5/(8*n*x**(8*n)) - 5*a**4*b/(7*n*x**(7*n)) - 5*a**3*b**2/(3*n*x**(6*n)) - 2*a**2*b**3/(n*x**(5*
n)) - 5*a*b**4/(4*n*x**(4*n)) - b**5/(3*n*x**(3*n)), Ne(n, 0)), ((a + b)**5*log(x), True))

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Giac [A]
time = 2.48, size = 74, normalized size = 0.96 \begin {gather*} -\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}} \end {gather*}

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))

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Mupad [B]
time = 1.36, size = 99, normalized size = 1.29 \begin {gather*} -\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}} \end {gather*}

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))

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