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]

-(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)
)

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Rubi [A]  time = 0.027085, antiderivative size = 77, normalized size of antiderivative = 1., 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 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]]

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 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]

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 = 0.031658, size = 74, normalized size = 0.96 \[ -\frac{x^{-8 n} \left (280 a^3 b^2 x^{2 n}+336 a^2 b^3 x^{3 n}+120 a^4 b x^n+21 a^5+210 a b^4 x^{4 n}+56 b^5 x^{5 n}\right )}{168 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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 = 0.018, size = 88, normalized size = 1.1 \begin{align*} -{\frac{{b}^{5}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{5\,a{b}^{4}}{4\,n \left ({x}^{n} \right ) ^{4}}}-2\,{\frac{{a}^{2}{b}^{3}}{n \left ({x}^{n} \right ) ^{5}}}-{\frac{5\,{a}^{3}{b}^{2}}{3\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{5\,{a}^{4}b}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{{a}^{5}}{8\,n \left ({x}^{n} \right ) ^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.38229, size = 170, normalized size = 2.21 \begin{align*} -\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{align*}

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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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Giac [A]  time = 1.19948, size = 100, normalized size = 1.3 \begin{align*} -\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{align*}

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))