3.2590 \(\int x^{-1-15 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=151 \[ -\frac{28 a^6 b^2 x^{-13 n}}{13 n}-\frac{14 a^5 b^3 x^{-12 n}}{3 n}-\frac{70 a^4 b^4 x^{-11 n}}{11 n}-\frac{28 a^3 b^5 x^{-10 n}}{5 n}-\frac{28 a^2 b^6 x^{-9 n}}{9 n}-\frac{4 a^7 b x^{-14 n}}{7 n}-\frac{a^8 x^{-15 n}}{15 n}-\frac{a b^7 x^{-8 n}}{n}-\frac{b^8 x^{-7 n}}{7 n} \]

[Out]

-a^8/(15*n*x^(15*n)) - (4*a^7*b)/(7*n*x^(14*n)) - (28*a^6*b^2)/(13*n*x^(13*n)) - (14*a^5*b^3)/(3*n*x^(12*n)) -
(70*a^4*b^4)/(11*n*x^(11*n)) - (28*a^3*b^5)/(5*n*x^(10*n)) - (28*a^2*b^6)/(9*n*x^(9*n)) - (a*b^7)/(n*x^(8*n))
- b^8/(7*n*x^(7*n))

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Rubi [A]  time = 0.0608719, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ -\frac{28 a^6 b^2 x^{-13 n}}{13 n}-\frac{14 a^5 b^3 x^{-12 n}}{3 n}-\frac{70 a^4 b^4 x^{-11 n}}{11 n}-\frac{28 a^3 b^5 x^{-10 n}}{5 n}-\frac{28 a^2 b^6 x^{-9 n}}{9 n}-\frac{4 a^7 b x^{-14 n}}{7 n}-\frac{a^8 x^{-15 n}}{15 n}-\frac{a b^7 x^{-8 n}}{n}-\frac{b^8 x^{-7 n}}{7 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 15*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(15*n*x^(15*n)) - (4*a^7*b)/(7*n*x^(14*n)) - (28*a^6*b^2)/(13*n*x^(13*n)) - (14*a^5*b^3)/(3*n*x^(12*n)) -
(70*a^4*b^4)/(11*n*x^(11*n)) - (28*a^3*b^5)/(5*n*x^(10*n)) - (28*a^2*b^6)/(9*n*x^(9*n)) - (a*b^7)/(n*x^(8*n))
- b^8/(7*n*x^(7*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^{-1-15 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{16}} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^8}{x^{16}}+\frac{8 a^7 b}{x^{15}}+\frac{28 a^6 b^2}{x^{14}}+\frac{56 a^5 b^3}{x^{13}}+\frac{70 a^4 b^4}{x^{12}}+\frac{56 a^3 b^5}{x^{11}}+\frac{28 a^2 b^6}{x^{10}}+\frac{8 a b^7}{x^9}+\frac{b^8}{x^8}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-15 n}}{15 n}-\frac{4 a^7 b x^{-14 n}}{7 n}-\frac{28 a^6 b^2 x^{-13 n}}{13 n}-\frac{14 a^5 b^3 x^{-12 n}}{3 n}-\frac{70 a^4 b^4 x^{-11 n}}{11 n}-\frac{28 a^3 b^5 x^{-10 n}}{5 n}-\frac{28 a^2 b^6 x^{-9 n}}{9 n}-\frac{a b^7 x^{-8 n}}{n}-\frac{b^8 x^{-7 n}}{7 n}\\ \end{align*}

Mathematica [A]  time = 0.0472497, size = 113, normalized size = 0.75 \[ -\frac{x^{-15 n} \left (97020 a^6 b^2 x^{2 n}+210210 a^5 b^3 x^{3 n}+286650 a^4 b^4 x^{4 n}+252252 a^3 b^5 x^{5 n}+140140 a^2 b^6 x^{6 n}+25740 a^7 b x^n+3003 a^8+45045 a b^7 x^{7 n}+6435 b^8 x^{8 n}\right )}{45045 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 15*n)*(a + b*x^n)^8,x]

[Out]

-(3003*a^8 + 25740*a^7*b*x^n + 97020*a^6*b^2*x^(2*n) + 210210*a^5*b^3*x^(3*n) + 286650*a^4*b^4*x^(4*n) + 25225
2*a^3*b^5*x^(5*n) + 140140*a^2*b^6*x^(6*n) + 45045*a*b^7*x^(7*n) + 6435*b^8*x^(8*n))/(45045*n*x^(15*n))

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Maple [A]  time = 0.023, size = 136, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{{b}^{7}a}{n \left ({x}^{n} \right ) ^{8}}}-{\frac{28\,{a}^{2}{b}^{6}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{28\,{a}^{3}{b}^{5}}{5\,n \left ({x}^{n} \right ) ^{10}}}-{\frac{70\,{a}^{4}{b}^{4}}{11\,n \left ({x}^{n} \right ) ^{11}}}-{\frac{14\,{a}^{5}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{12}}}-{\frac{28\,{a}^{6}{b}^{2}}{13\,n \left ({x}^{n} \right ) ^{13}}}-{\frac{4\,b{a}^{7}}{7\,n \left ({x}^{n} \right ) ^{14}}}-{\frac{{a}^{8}}{15\,n \left ({x}^{n} \right ) ^{15}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-15*n)*(a+b*x^n)^8,x)

[Out]

-1/7*b^8/n/(x^n)^7-a*b^7/n/(x^n)^8-28/9*a^2*b^6/n/(x^n)^9-28/5*a^3*b^5/n/(x^n)^10-70/11*a^4*b^4/n/(x^n)^11-14/
3*a^5*b^3/n/(x^n)^12-28/13*a^6*b^2/n/(x^n)^13-4/7*a^7*b/n/(x^n)^14-1/15*a^8/n/(x^n)^15

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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^(-1-15*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.08145, size = 293, normalized size = 1.94 \begin{align*} -\frac{6435 \, b^{8} x^{8 \, n} + 45045 \, a b^{7} x^{7 \, n} + 140140 \, a^{2} b^{6} x^{6 \, n} + 252252 \, a^{3} b^{5} x^{5 \, n} + 286650 \, a^{4} b^{4} x^{4 \, n} + 210210 \, a^{5} b^{3} x^{3 \, n} + 97020 \, a^{6} b^{2} x^{2 \, n} + 25740 \, a^{7} b x^{n} + 3003 \, a^{8}}{45045 \, n x^{15 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-15*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

-1/45045*(6435*b^8*x^(8*n) + 45045*a*b^7*x^(7*n) + 140140*a^2*b^6*x^(6*n) + 252252*a^3*b^5*x^(5*n) + 286650*a^
4*b^4*x^(4*n) + 210210*a^5*b^3*x^(3*n) + 97020*a^6*b^2*x^(2*n) + 25740*a^7*b*x^n + 3003*a^8)/(n*x^(15*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**(-1-15*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.24264, size = 153, normalized size = 1.01 \begin{align*} -\frac{6435 \, b^{8} x^{8 \, n} + 45045 \, a b^{7} x^{7 \, n} + 140140 \, a^{2} b^{6} x^{6 \, n} + 252252 \, a^{3} b^{5} x^{5 \, n} + 286650 \, a^{4} b^{4} x^{4 \, n} + 210210 \, a^{5} b^{3} x^{3 \, n} + 97020 \, a^{6} b^{2} x^{2 \, n} + 25740 \, a^{7} b x^{n} + 3003 \, a^{8}}{45045 \, n x^{15 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-15*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

-1/45045*(6435*b^8*x^(8*n) + 45045*a*b^7*x^(7*n) + 140140*a^2*b^6*x^(6*n) + 252252*a^3*b^5*x^(5*n) + 286650*a^
4*b^4*x^(4*n) + 210210*a^5*b^3*x^(3*n) + 97020*a^6*b^2*x^(2*n) + 25740*a^7*b*x^n + 3003*a^8)/(n*x^(15*n))