3.2567 \(\int x^{-1+8 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=151 \[ \frac{14 a^6 b^2 x^{10 n}}{5 n}+\frac{56 a^5 b^3 x^{11 n}}{11 n}+\frac{35 a^4 b^4 x^{12 n}}{6 n}+\frac{56 a^3 b^5 x^{13 n}}{13 n}+\frac{2 a^2 b^6 x^{14 n}}{n}+\frac{8 a^7 b x^{9 n}}{9 n}+\frac{a^8 x^{8 n}}{8 n}+\frac{8 a b^7 x^{15 n}}{15 n}+\frac{b^8 x^{16 n}}{16 n} \]

[Out]

(a^8*x^(8*n))/(8*n) + (8*a^7*b*x^(9*n))/(9*n) + (14*a^6*b^2*x^(10*n))/(5*n) + (56*a^5*b^3*x^(11*n))/(11*n) + (
35*a^4*b^4*x^(12*n))/(6*n) + (56*a^3*b^5*x^(13*n))/(13*n) + (2*a^2*b^6*x^(14*n))/n + (8*a*b^7*x^(15*n))/(15*n)
+ (b^8*x^(16*n))/(16*n)

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Rubi [A]  time = 0.0721068, 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} \[ \frac{14 a^6 b^2 x^{10 n}}{5 n}+\frac{56 a^5 b^3 x^{11 n}}{11 n}+\frac{35 a^4 b^4 x^{12 n}}{6 n}+\frac{56 a^3 b^5 x^{13 n}}{13 n}+\frac{2 a^2 b^6 x^{14 n}}{n}+\frac{8 a^7 b x^{9 n}}{9 n}+\frac{a^8 x^{8 n}}{8 n}+\frac{8 a b^7 x^{15 n}}{15 n}+\frac{b^8 x^{16 n}}{16 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 8*n)*(a + b*x^n)^8,x]

[Out]

(a^8*x^(8*n))/(8*n) + (8*a^7*b*x^(9*n))/(9*n) + (14*a^6*b^2*x^(10*n))/(5*n) + (56*a^5*b^3*x^(11*n))/(11*n) + (
35*a^4*b^4*x^(12*n))/(6*n) + (56*a^3*b^5*x^(13*n))/(13*n) + (2*a^2*b^6*x^(14*n))/n + (8*a*b^7*x^(15*n))/(15*n)
+ (b^8*x^(16*n))/(16*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+8 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int x^7 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^8 x^7+8 a^7 b x^8+28 a^6 b^2 x^9+56 a^5 b^3 x^{10}+70 a^4 b^4 x^{11}+56 a^3 b^5 x^{12}+28 a^2 b^6 x^{13}+8 a b^7 x^{14}+b^8 x^{15}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{a^8 x^{8 n}}{8 n}+\frac{8 a^7 b x^{9 n}}{9 n}+\frac{14 a^6 b^2 x^{10 n}}{5 n}+\frac{56 a^5 b^3 x^{11 n}}{11 n}+\frac{35 a^4 b^4 x^{12 n}}{6 n}+\frac{56 a^3 b^5 x^{13 n}}{13 n}+\frac{2 a^2 b^6 x^{14 n}}{n}+\frac{8 a b^7 x^{15 n}}{15 n}+\frac{b^8 x^{16 n}}{16 n}\\ \end{align*}

Mathematica [A]  time = 0.056654, size = 128, normalized size = 0.85 \[ \frac{\frac{14}{5} a^6 b^2 x^{10 n}+\frac{56}{11} a^5 b^3 x^{11 n}+\frac{35}{6} a^4 b^4 x^{12 n}+\frac{56}{13} a^3 b^5 x^{13 n}+2 a^2 b^6 x^{14 n}+\frac{8}{9} a^7 b x^{9 n}+\frac{1}{8} a^8 x^{8 n}+\frac{8}{15} a b^7 x^{15 n}+\frac{1}{16} b^8 x^{16 n}}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^8*x^(8*n))/8 + (8*a^7*b*x^(9*n))/9 + (14*a^6*b^2*x^(10*n))/5 + (56*a^5*b^3*x^(11*n))/11 + (35*a^4*b^4*x^(1
2*n))/6 + (56*a^3*b^5*x^(13*n))/13 + 2*a^2*b^6*x^(14*n) + (8*a*b^7*x^(15*n))/15 + (b^8*x^(16*n))/16)/n

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Maple [A]  time = 0.021, size = 136, normalized size = 0.9 \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{16}}{16\,n}}+{\frac{8\,{b}^{7}a \left ({x}^{n} \right ) ^{15}}{15\,n}}+2\,{\frac{{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{14}}{n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{13}}{13\,n}}+{\frac{35\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{12}}{6\,n}}+{\frac{56\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{14\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{8\,b{a}^{7} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{8}}{8\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*b^8/n*(x^n)^16+8/15*a*b^7/n*(x^n)^15+2*a^2*b^6/n*(x^n)^14+56/13*a^3*b^5/n*(x^n)^13+35/6*a^4*b^4/n*(x^n)^1
2+56/11*a^5*b^3/n*(x^n)^11+14/5*a^6*b^2/n*(x^n)^10+8/9*a^7*b/n*(x^n)^9+1/8*a^8/n*(x^n)^8

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33022, size = 306, normalized size = 2.03 \begin{align*} \frac{6435 \, b^{8} x^{16 \, n} + 54912 \, a b^{7} x^{15 \, n} + 205920 \, a^{2} b^{6} x^{14 \, n} + 443520 \, a^{3} b^{5} x^{13 \, n} + 600600 \, a^{4} b^{4} x^{12 \, n} + 524160 \, a^{5} b^{3} x^{11 \, n} + 288288 \, a^{6} b^{2} x^{10 \, n} + 91520 \, a^{7} b x^{9 \, n} + 12870 \, a^{8} x^{8 \, n}}{102960 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/102960*(6435*b^8*x^(16*n) + 54912*a*b^7*x^(15*n) + 205920*a^2*b^6*x^(14*n) + 443520*a^3*b^5*x^(13*n) + 60060
0*a^4*b^4*x^(12*n) + 524160*a^5*b^3*x^(11*n) + 288288*a^6*b^2*x^(10*n) + 91520*a^7*b*x^(9*n) + 12870*a^8*x^(8*
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+8*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^{8 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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