3.2586 \(\int x^{-1-11 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=77 \[ -\frac{b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n}+\frac{b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac{x^{-11 n} \left (a+b x^n\right )^9}{11 a n} \]

[Out]

-(a + b*x^n)^9/(11*a*n*x^(11*n)) + (b*(a + b*x^n)^9)/(55*a^2*n*x^(10*n)) - (b^2*(a + b*x^n)^9)/(495*a^3*n*x^(9
*n))

________________________________________________________________________________________

Rubi [A]  time = 0.0290084, 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} \[ -\frac{b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n}+\frac{b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac{x^{-11 n} \left (a+b x^n\right )^9}{11 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 11*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(11*a*n*x^(11*n)) + (b*(a + b*x^n)^9)/(55*a^2*n*x^(10*n)) - (b^2*(a + b*x^n)^9)/(495*a^3*n*x^(9
*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^{-1-11 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{12}} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-11 n} \left (a+b x^n\right )^9}{11 a n}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^n\right )}{11 a n}\\ &=-\frac{x^{-11 n} \left (a+b x^n\right )^9}{11 a n}+\frac{b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^n\right )}{55 a^2 n}\\ &=-\frac{x^{-11 n} \left (a+b x^n\right )^9}{11 a n}+\frac{b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac{b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n}\\ \end{align*}

Mathematica [A]  time = 0.0156277, size = 46, normalized size = 0.6 \[ -\frac{x^{-11 n} \left (a+b x^n\right )^9 \left (45 a^2-9 a b x^n+b^2 x^{2 n}\right )}{495 a^3 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 11*n)*(a + b*x^n)^8,x]

[Out]

-((a + b*x^n)^9*(45*a^2 - 9*a*b*x^n + b^2*x^(2*n)))/(495*a^3*n*x^(11*n))

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 136, normalized size = 1.8 \begin{align*} -{\frac{{b}^{8}}{3\,n \left ({x}^{n} \right ) ^{3}}}-2\,{\frac{{b}^{7}a}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{28\,{a}^{2}{b}^{6}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{28\,{a}^{3}{b}^{5}}{3\,n \left ({x}^{n} \right ) ^{6}}}-10\,{\frac{{a}^{4}{b}^{4}}{n \left ({x}^{n} \right ) ^{7}}}-7\,{\frac{{a}^{5}{b}^{3}}{n \left ({x}^{n} \right ) ^{8}}}-{\frac{28\,{a}^{6}{b}^{2}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{4\,b{a}^{7}}{5\,n \left ({x}^{n} \right ) ^{10}}}-{\frac{{a}^{8}}{11\,n \left ({x}^{n} \right ) ^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-11*n)*(a+b*x^n)^8,x)

[Out]

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

________________________________________________________________________________________

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^(-1-11*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.05674, size = 269, normalized size = 3.49 \begin{align*} -\frac{165 \, b^{8} x^{8 \, n} + 990 \, a b^{7} x^{7 \, n} + 2772 \, a^{2} b^{6} x^{6 \, n} + 4620 \, a^{3} b^{5} x^{5 \, n} + 4950 \, a^{4} b^{4} x^{4 \, n} + 3465 \, a^{5} b^{3} x^{3 \, n} + 1540 \, a^{6} b^{2} x^{2 \, n} + 396 \, a^{7} b x^{n} + 45 \, a^{8}}{495 \, n x^{11 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-11*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

-1/495*(165*b^8*x^(8*n) + 990*a*b^7*x^(7*n) + 2772*a^2*b^6*x^(6*n) + 4620*a^3*b^5*x^(5*n) + 4950*a^4*b^4*x^(4*
n) + 3465*a^5*b^3*x^(3*n) + 1540*a^6*b^2*x^(2*n) + 396*a^7*b*x^n + 45*a^8)/(n*x^(11*n))

________________________________________________________________________________________

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**(-1-11*n)*(a+b*x**n)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.27122, size = 153, normalized size = 1.99 \begin{align*} -\frac{165 \, b^{8} x^{8 \, n} + 990 \, a b^{7} x^{7 \, n} + 2772 \, a^{2} b^{6} x^{6 \, n} + 4620 \, a^{3} b^{5} x^{5 \, n} + 4950 \, a^{4} b^{4} x^{4 \, n} + 3465 \, a^{5} b^{3} x^{3 \, n} + 1540 \, a^{6} b^{2} x^{2 \, n} + 396 \, a^{7} b x^{n} + 45 \, a^{8}}{495 \, n x^{11 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-11*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

-1/495*(165*b^8*x^(8*n) + 990*a*b^7*x^(7*n) + 2772*a^2*b^6*x^(6*n) + 4620*a^3*b^5*x^(5*n) + 4950*a^4*b^4*x^(4*
n) + 3465*a^5*b^3*x^(3*n) + 1540*a^6*b^2*x^(2*n) + 396*a^7*b*x^n + 45*a^8)/(n*x^(11*n))