3.2566 \(\int x^{-1+9 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=151 \[ \frac{28 a^6 b^2 x^{11 n}}{11 n}+\frac{14 a^5 b^3 x^{12 n}}{3 n}+\frac{70 a^4 b^4 x^{13 n}}{13 n}+\frac{4 a^3 b^5 x^{14 n}}{n}+\frac{28 a^2 b^6 x^{15 n}}{15 n}+\frac{4 a^7 b x^{10 n}}{5 n}+\frac{a^8 x^{9 n}}{9 n}+\frac{a b^7 x^{16 n}}{2 n}+\frac{b^8 x^{17 n}}{17 n} \]

[Out]

(a^8*x^(9*n))/(9*n) + (4*a^7*b*x^(10*n))/(5*n) + (28*a^6*b^2*x^(11*n))/(11*n) + (14*a^5*b^3*x^(12*n))/(3*n) +
(70*a^4*b^4*x^(13*n))/(13*n) + (4*a^3*b^5*x^(14*n))/n + (28*a^2*b^6*x^(15*n))/(15*n) + (a*b^7*x^(16*n))/(2*n)
+ (b^8*x^(17*n))/(17*n)

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Rubi [A]  time = 0.0783979, 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} \[ \frac{28 a^6 b^2 x^{11 n}}{11 n}+\frac{14 a^5 b^3 x^{12 n}}{3 n}+\frac{70 a^4 b^4 x^{13 n}}{13 n}+\frac{4 a^3 b^5 x^{14 n}}{n}+\frac{28 a^2 b^6 x^{15 n}}{15 n}+\frac{4 a^7 b x^{10 n}}{5 n}+\frac{a^8 x^{9 n}}{9 n}+\frac{a b^7 x^{16 n}}{2 n}+\frac{b^8 x^{17 n}}{17 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 9*n)*(a + b*x^n)^8,x]

[Out]

(a^8*x^(9*n))/(9*n) + (4*a^7*b*x^(10*n))/(5*n) + (28*a^6*b^2*x^(11*n))/(11*n) + (14*a^5*b^3*x^(12*n))/(3*n) +
(70*a^4*b^4*x^(13*n))/(13*n) + (4*a^3*b^5*x^(14*n))/n + (28*a^2*b^6*x^(15*n))/(15*n) + (a*b^7*x^(16*n))/(2*n)
+ (b^8*x^(17*n))/(17*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+9 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int x^8 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^8 x^8+8 a^7 b x^9+28 a^6 b^2 x^{10}+56 a^5 b^3 x^{11}+70 a^4 b^4 x^{12}+56 a^3 b^5 x^{13}+28 a^2 b^6 x^{14}+8 a b^7 x^{15}+b^8 x^{16}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{a^8 x^{9 n}}{9 n}+\frac{4 a^7 b x^{10 n}}{5 n}+\frac{28 a^6 b^2 x^{11 n}}{11 n}+\frac{14 a^5 b^3 x^{12 n}}{3 n}+\frac{70 a^4 b^4 x^{13 n}}{13 n}+\frac{4 a^3 b^5 x^{14 n}}{n}+\frac{28 a^2 b^6 x^{15 n}}{15 n}+\frac{a b^7 x^{16 n}}{2 n}+\frac{b^8 x^{17 n}}{17 n}\\ \end{align*}

Mathematica [A]  time = 0.0688217, size = 128, normalized size = 0.85 \[ \frac{\frac{28}{11} a^6 b^2 x^{11 n}+\frac{14}{3} a^5 b^3 x^{12 n}+\frac{70}{13} a^4 b^4 x^{13 n}+4 a^3 b^5 x^{14 n}+\frac{28}{15} a^2 b^6 x^{15 n}+\frac{4}{5} a^7 b x^{10 n}+\frac{1}{9} a^8 x^{9 n}+\frac{1}{2} a b^7 x^{16 n}+\frac{1}{17} b^8 x^{17 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 9*n)*(a + b*x^n)^8,x]

[Out]

((a^8*x^(9*n))/9 + (4*a^7*b*x^(10*n))/5 + (28*a^6*b^2*x^(11*n))/11 + (14*a^5*b^3*x^(12*n))/3 + (70*a^4*b^4*x^(
13*n))/13 + 4*a^3*b^5*x^(14*n) + (28*a^2*b^6*x^(15*n))/15 + (a*b^7*x^(16*n))/2 + (b^8*x^(17*n))/17)/n

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Maple [A]  time = 0.023, size = 136, normalized size = 0.9 \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{17}}{17\,n}}+{\frac{{b}^{7}a \left ({x}^{n} \right ) ^{16}}{2\,n}}+{\frac{28\,{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{15}}{15\,n}}+4\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{14}}{n}}+{\frac{70\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{13}}{13\,n}}+{\frac{14\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{12}}{3\,n}}+{\frac{28\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{4\,b{a}^{7} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{9}}{9\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+9*n)*(a+b*x^n)^8,x)

[Out]

1/17*b^8/n*(x^n)^17+1/2*a*b^7/n*(x^n)^16+28/15*a^2*b^6/n*(x^n)^15+4*a^3*b^5/n*(x^n)^14+70/13*a^4*b^4/n*(x^n)^1
3+14/3*a^5*b^3/n*(x^n)^12+28/11*a^6*b^2/n*(x^n)^11+4/5*a^7*b/n*(x^n)^10+1/9*a^8/n*(x^n)^9

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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+9*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.30743, size = 315, normalized size = 2.09 \begin{align*} \frac{12870 \, b^{8} x^{17 \, n} + 109395 \, a b^{7} x^{16 \, n} + 408408 \, a^{2} b^{6} x^{15 \, n} + 875160 \, a^{3} b^{5} x^{14 \, n} + 1178100 \, a^{4} b^{4} x^{13 \, n} + 1021020 \, a^{5} b^{3} x^{12 \, n} + 556920 \, a^{6} b^{2} x^{11 \, n} + 175032 \, a^{7} b x^{10 \, n} + 24310 \, a^{8} x^{9 \, n}}{218790 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+9*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/218790*(12870*b^8*x^(17*n) + 109395*a*b^7*x^(16*n) + 408408*a^2*b^6*x^(15*n) + 875160*a^3*b^5*x^(14*n) + 117
8100*a^4*b^4*x^(13*n) + 1021020*a^5*b^3*x^(12*n) + 556920*a^6*b^2*x^(11*n) + 175032*a^7*b*x^(10*n) + 24310*a^8
*x^(9*n))/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+9*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{8} x^{9 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+9*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(9*n - 1), x)