3.2585 \(\int x^{-1-10 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=50 \[ \frac{b x^{-9 n} \left (a+b x^n\right )^9}{90 a^2 n}-\frac{x^{-10 n} \left (a+b x^n\right )^9}{10 a n} \]

[Out]

-(a + b*x^n)^9/(10*a*n*x^(10*n)) + (b*(a + b*x^n)^9)/(90*a^2*n*x^(9*n))

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Rubi [A]  time = 0.0191083, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {266, 45, 37} \[ \frac{b x^{-9 n} \left (a+b x^n\right )^9}{90 a^2 n}-\frac{x^{-10 n} \left (a+b x^n\right )^9}{10 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 10*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(10*a*n*x^(10*n)) + (b*(a + b*x^n)^9)/(90*a^2*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^{-1-10 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-10 n} \left (a+b x^n\right )^9}{10 a n}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^n\right )}{10 a n}\\ &=-\frac{x^{-10 n} \left (a+b x^n\right )^9}{10 a n}+\frac{b x^{-9 n} \left (a+b x^n\right )^9}{90 a^2 n}\\ \end{align*}

Mathematica [A]  time = 0.013021, size = 33, normalized size = 0.66 \[ \frac{x^{-10 n} \left (b x^n-9 a\right ) \left (a+b x^n\right )^9}{90 a^2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 10*n)*(a + b*x^n)^8,x]

[Out]

((-9*a + b*x^n)*(a + b*x^n)^9)/(90*a^2*n*x^(10*n))

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Maple [B]  time = 0.023, size = 136, normalized size = 2.7 \begin{align*} -{\frac{{b}^{8}}{2\,n \left ({x}^{n} \right ) ^{2}}}-{\frac{8\,{b}^{7}a}{3\,n \left ({x}^{n} \right ) ^{3}}}-7\,{\frac{{a}^{2}{b}^{6}}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{56\,{a}^{3}{b}^{5}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{35\,{a}^{4}{b}^{4}}{3\,n \left ({x}^{n} \right ) ^{6}}}-8\,{\frac{{a}^{5}{b}^{3}}{n \left ({x}^{n} \right ) ^{7}}}-{\frac{7\,{a}^{6}{b}^{2}}{2\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{8\,b{a}^{7}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{{a}^{8}}{10\,n \left ({x}^{n} \right ) ^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-10*n)*(a+b*x^n)^8,x)

[Out]

-1/2*b^8/n/(x^n)^2-8/3*a*b^7/n/(x^n)^3-7*a^2*b^6/n/(x^n)^4-56/5*a^3*b^5/n/(x^n)^5-35/3*a^4*b^4/n/(x^n)^6-8*a^5
*b^3/n/(x^n)^7-7/2*a^6*b^2/n/(x^n)^8-8/9*a^7*b/n/(x^n)^9-1/10*a^8/n/(x^n)^10

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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^(-1-10*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.06096, size = 259, normalized size = 5.18 \begin{align*} -\frac{45 \, b^{8} x^{8 \, n} + 240 \, a b^{7} x^{7 \, n} + 630 \, a^{2} b^{6} x^{6 \, n} + 1008 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 720 \, a^{5} b^{3} x^{3 \, n} + 315 \, a^{6} b^{2} x^{2 \, n} + 80 \, a^{7} b x^{n} + 9 \, a^{8}}{90 \, n x^{10 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-10*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

-1/90*(45*b^8*x^(8*n) + 240*a*b^7*x^(7*n) + 630*a^2*b^6*x^(6*n) + 1008*a^3*b^5*x^(5*n) + 1050*a^4*b^4*x^(4*n)
+ 720*a^5*b^3*x^(3*n) + 315*a^6*b^2*x^(2*n) + 80*a^7*b*x^n + 9*a^8)/(n*x^(10*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**(-1-10*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.27476, size = 153, normalized size = 3.06 \begin{align*} -\frac{45 \, b^{8} x^{8 \, n} + 240 \, a b^{7} x^{7 \, n} + 630 \, a^{2} b^{6} x^{6 \, n} + 1008 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 720 \, a^{5} b^{3} x^{3 \, n} + 315 \, a^{6} b^{2} x^{2 \, n} + 80 \, a^{7} b x^{n} + 9 \, a^{8}}{90 \, n x^{10 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-10*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

-1/90*(45*b^8*x^(8*n) + 240*a*b^7*x^(7*n) + 630*a^2*b^6*x^(6*n) + 1008*a^3*b^5*x^(5*n) + 1050*a^4*b^4*x^(4*n)
+ 720*a^5*b^3*x^(3*n) + 315*a^6*b^2*x^(2*n) + 80*a^7*b*x^n + 9*a^8)/(n*x^(10*n))