3.2587 \(\int x^{-1-12 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=104 \[ -\frac{b^2 x^{-10 n} \left (a+b x^n\right )^9}{220 a^3 n}+\frac{b^3 x^{-9 n} \left (a+b x^n\right )^9}{1980 a^4 n}+\frac{b x^{-11 n} \left (a+b x^n\right )^9}{44 a^2 n}-\frac{x^{-12 n} \left (a+b x^n\right )^9}{12 a n} \]

[Out]

-(a + b*x^n)^9/(12*a*n*x^(12*n)) + (b*(a + b*x^n)^9)/(44*a^2*n*x^(11*n)) - (b^2*(a + b*x^n)^9)/(220*a^3*n*x^(1
0*n)) + (b^3*(a + b*x^n)^9)/(1980*a^4*n*x^(9*n))

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Rubi [A]  time = 0.0395928, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {266, 45, 37} \[ -\frac{b^2 x^{-10 n} \left (a+b x^n\right )^9}{220 a^3 n}+\frac{b^3 x^{-9 n} \left (a+b x^n\right )^9}{1980 a^4 n}+\frac{b x^{-11 n} \left (a+b x^n\right )^9}{44 a^2 n}-\frac{x^{-12 n} \left (a+b x^n\right )^9}{12 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 12*n)*(a + b*x^n)^8,x]

[Out]

-(a + b*x^n)^9/(12*a*n*x^(12*n)) + (b*(a + b*x^n)^9)/(44*a^2*n*x^(11*n)) - (b^2*(a + b*x^n)^9)/(220*a^3*n*x^(1
0*n)) + (b^3*(a + b*x^n)^9)/(1980*a^4*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^{-1-12 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{13}} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-12 n} \left (a+b x^n\right )^9}{12 a n}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{12}} \, dx,x,x^n\right )}{4 a n}\\ &=-\frac{x^{-12 n} \left (a+b x^n\right )^9}{12 a n}+\frac{b x^{-11 n} \left (a+b x^n\right )^9}{44 a^2 n}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^n\right )}{22 a^2 n}\\ &=-\frac{x^{-12 n} \left (a+b x^n\right )^9}{12 a n}+\frac{b x^{-11 n} \left (a+b x^n\right )^9}{44 a^2 n}-\frac{b^2 x^{-10 n} \left (a+b x^n\right )^9}{220 a^3 n}-\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^n\right )}{220 a^3 n}\\ &=-\frac{x^{-12 n} \left (a+b x^n\right )^9}{12 a n}+\frac{b x^{-11 n} \left (a+b x^n\right )^9}{44 a^2 n}-\frac{b^2 x^{-10 n} \left (a+b x^n\right )^9}{220 a^3 n}+\frac{b^3 x^{-9 n} \left (a+b x^n\right )^9}{1980 a^4 n}\\ \end{align*}

Mathematica [A]  time = 0.0185106, size = 59, normalized size = 0.57 \[ \frac{x^{-12 n} \left (a+b x^n\right )^9 \left (45 a^2 b x^n-165 a^3-9 a b^2 x^{2 n}+b^3 x^{3 n}\right )}{1980 a^4 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 12*n)*(a + b*x^n)^8,x]

[Out]

((a + b*x^n)^9*(-165*a^3 + 45*a^2*b*x^n - 9*a*b^2*x^(2*n) + b^3*x^(3*n)))/(1980*a^4*n*x^(12*n))

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Maple [A]  time = 0.022, size = 136, normalized size = 1.3 \begin{align*} -{\frac{{b}^{8}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{8\,{b}^{7}a}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{14\,{a}^{2}{b}^{6}}{3\,n \left ({x}^{n} \right ) ^{6}}}-8\,{\frac{{a}^{3}{b}^{5}}{n \left ({x}^{n} \right ) ^{7}}}-{\frac{35\,{a}^{4}{b}^{4}}{4\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{56\,{a}^{5}{b}^{3}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{14\,{a}^{6}{b}^{2}}{5\,n \left ({x}^{n} \right ) ^{10}}}-{\frac{8\,b{a}^{7}}{11\,n \left ({x}^{n} \right ) ^{11}}}-{\frac{{a}^{8}}{12\,n \left ({x}^{n} \right ) ^{12}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-12*n)*(a+b*x^n)^8,x)

[Out]

-1/4*b^8/n/(x^n)^4-8/5*a*b^7/n/(x^n)^5-14/3*a^2*b^6/n/(x^n)^6-8*a^3*b^5/n/(x^n)^7-35/4*a^4*b^4/n/(x^n)^8-56/9*
a^5*b^3/n/(x^n)^9-14/5*a^6*b^2/n/(x^n)^10-8/11*a^7*b/n/(x^n)^11-1/12*a^8/n/(x^n)^12

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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^(-1-12*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.06772, size = 278, normalized size = 2.67 \begin{align*} -\frac{495 \, b^{8} x^{8 \, n} + 3168 \, a b^{7} x^{7 \, n} + 9240 \, a^{2} b^{6} x^{6 \, n} + 15840 \, a^{3} b^{5} x^{5 \, n} + 17325 \, a^{4} b^{4} x^{4 \, n} + 12320 \, a^{5} b^{3} x^{3 \, n} + 5544 \, a^{6} b^{2} x^{2 \, n} + 1440 \, a^{7} b x^{n} + 165 \, a^{8}}{1980 \, n x^{12 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-12*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

-1/1980*(495*b^8*x^(8*n) + 3168*a*b^7*x^(7*n) + 9240*a^2*b^6*x^(6*n) + 15840*a^3*b^5*x^(5*n) + 17325*a^4*b^4*x
^(4*n) + 12320*a^5*b^3*x^(3*n) + 5544*a^6*b^2*x^(2*n) + 1440*a^7*b*x^n + 165*a^8)/(n*x^(12*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**(-1-12*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.23943, size = 153, normalized size = 1.47 \begin{align*} -\frac{495 \, b^{8} x^{8 \, n} + 3168 \, a b^{7} x^{7 \, n} + 9240 \, a^{2} b^{6} x^{6 \, n} + 15840 \, a^{3} b^{5} x^{5 \, n} + 17325 \, a^{4} b^{4} x^{4 \, n} + 12320 \, a^{5} b^{3} x^{3 \, n} + 5544 \, a^{6} b^{2} x^{2 \, n} + 1440 \, a^{7} b x^{n} + 165 \, a^{8}}{1980 \, n x^{12 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-12*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

-1/1980*(495*b^8*x^(8*n) + 3168*a*b^7*x^(7*n) + 9240*a^2*b^6*x^(6*n) + 15840*a^3*b^5*x^(5*n) + 17325*a^4*b^4*x
^(4*n) + 12320*a^5*b^3*x^(3*n) + 5544*a^6*b^2*x^(2*n) + 1440*a^7*b*x^n + 165*a^8)/(n*x^(12*n))