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

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

[Out]

-1/12*(a+b*x^n)^9/a/n/(x^(12*n))+1/44*b*(a+b*x^n)^9/a^2/n/(x^(11*n))-1/220*b^2*(a+b*x^n)^9/a^3/n/(x^(10*n))+1/
1980*b^3*(a+b*x^n)^9/a^4/n/(x^(9*n))

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Rubi [A]  time = 0.04, antiderivative size = 104, normalized size of antiderivative = 1.00, 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 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 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 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]]

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*}

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Mathematica [A]  time = 0.02, size = 59, normalized size = 0.57 \[ \frac {x^{-12 n} \left (a+b x^n\right )^9 \left (-165 a^3+45 a^2 b x^n-9 a b^2 x^{2 n}+b^3 x^{3 n}\right )}{1980 a^4 n} \]

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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fricas [A]  time = 0.60, size = 113, normalized size = 1.09 \[ -\frac {495 \, b^{8} x^{8 \, n} + 3168 \, a b^{7} x^{7 \, n} + 9240 \, a^{2} b^{6} x^{6 \, n} + 15840 \, a^{3} b^{5} x^{5 \, n} + 17325 \, a^{4} b^{4} x^{4 \, n} + 12320 \, a^{5} b^{3} x^{3 \, n} + 5544 \, a^{6} b^{2} x^{2 \, n} + 1440 \, a^{7} b x^{n} + 165 \, a^{8}}{1980 \, n x^{12 \, n}} \]

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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giac [A]  time = 0.45, size = 113, normalized size = 1.09 \[ -\frac {495 \, b^{8} x^{8 \, n} + 3168 \, a b^{7} x^{7 \, n} + 9240 \, a^{2} b^{6} x^{6 \, n} + 15840 \, a^{3} b^{5} x^{5 \, n} + 17325 \, a^{4} b^{4} x^{4 \, n} + 12320 \, a^{5} b^{3} x^{3 \, n} + 5544 \, a^{6} b^{2} x^{2 \, n} + 1440 \, a^{7} b x^{n} + 165 \, a^{8}}{1980 \, n x^{12 \, n}} \]

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))

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maple [A]  time = 0.03, size = 136, normalized size = 1.31 \[ -\frac {a^{8} x^{-12 n}}{12 n}-\frac {8 a^{7} b \,x^{-11 n}}{11 n}-\frac {14 a^{6} b^{2} x^{-10 n}}{5 n}-\frac {56 a^{5} b^{3} x^{-9 n}}{9 n}-\frac {35 a^{4} b^{4} x^{-8 n}}{4 n}-\frac {8 a^{3} b^{5} x^{-7 n}}{n}-\frac {14 a^{2} b^{6} x^{-6 n}}{3 n}-\frac {8 a \,b^{7} x^{-5 n}}{5 n}-\frac {b^{8} x^{-4 n}}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-12*n)*(b*x^n+a)^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 [A]  time = 0.53, size = 153, normalized size = 1.47 \[ -\frac {a^{8}}{12 \, n x^{12 \, n}} - \frac {8 \, a^{7} b}{11 \, n x^{11 \, n}} - \frac {14 \, a^{6} b^{2}}{5 \, n x^{10 \, n}} - \frac {56 \, a^{5} b^{3}}{9 \, n x^{9 \, n}} - \frac {35 \, a^{4} b^{4}}{4 \, n x^{8 \, n}} - \frac {8 \, a^{3} b^{5}}{n x^{7 \, n}} - \frac {14 \, a^{2} b^{6}}{3 \, n x^{6 \, n}} - \frac {8 \, a b^{7}}{5 \, n x^{5 \, n}} - \frac {b^{8}}{4 \, n x^{4 \, n}} \]

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]

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

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mupad [B]  time = 1.43, size = 153, normalized size = 1.47 \[ -\frac {a^8}{12\,n\,x^{12\,n}}-\frac {b^8}{4\,n\,x^{4\,n}}-\frac {14\,a^2\,b^6}{3\,n\,x^{6\,n}}-\frac {8\,a^3\,b^5}{n\,x^{7\,n}}-\frac {35\,a^4\,b^4}{4\,n\,x^{8\,n}}-\frac {56\,a^5\,b^3}{9\,n\,x^{9\,n}}-\frac {14\,a^6\,b^2}{5\,n\,x^{10\,n}}-\frac {8\,a\,b^7}{5\,n\,x^{5\,n}}-\frac {8\,a^7\,b}{11\,n\,x^{11\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^8/x^(12*n + 1),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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