3.2570 \(\int x^{-1+5 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=106 \[ \frac{6 a^2 \left (a+b x^n\right )^{11}}{11 b^5 n}-\frac{2 a^3 \left (a+b x^n\right )^{10}}{5 b^5 n}+\frac{a^4 \left (a+b x^n\right )^9}{9 b^5 n}+\frac{\left (a+b x^n\right )^{13}}{13 b^5 n}-\frac{a \left (a+b x^n\right )^{12}}{3 b^5 n} \]

[Out]

(a^4*(a + b*x^n)^9)/(9*b^5*n) - (2*a^3*(a + b*x^n)^10)/(5*b^5*n) + (6*a^2*(a + b*x^n)^11)/(11*b^5*n) - (a*(a +
b*x^n)^12)/(3*b^5*n) + (a + b*x^n)^13/(13*b^5*n)

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Rubi [A]  time = 0.0583547, antiderivative size = 106, 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} \[ \frac{6 a^2 \left (a+b x^n\right )^{11}}{11 b^5 n}-\frac{2 a^3 \left (a+b x^n\right )^{10}}{5 b^5 n}+\frac{a^4 \left (a+b x^n\right )^9}{9 b^5 n}+\frac{\left (a+b x^n\right )^{13}}{13 b^5 n}-\frac{a \left (a+b x^n\right )^{12}}{3 b^5 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 5*n)*(a + b*x^n)^8,x]

[Out]

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

Mathematica [A]  time = 0.043355, size = 66, normalized size = 0.62 \[ \frac{\left (a+b x^n\right )^9 \left (45 a^2 b^2 x^{2 n}-9 a^3 b x^n+a^4-165 a b^3 x^{3 n}+495 b^4 x^{4 n}\right )}{6435 b^5 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 5*n)*(a + b*x^n)^8,x]

[Out]

((a + b*x^n)^9*(a^4 - 9*a^3*b*x^n + 45*a^2*b^2*x^(2*n) - 165*a*b^3*x^(3*n) + 495*b^4*x^(4*n)))/(6435*b^5*n)

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Maple [A]  time = 0.022, size = 136, normalized size = 1.3 \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{13}}{13\,n}}+{\frac{2\,{b}^{7}a \left ({x}^{n} \right ) ^{12}}{3\,n}}+{\frac{28\,{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{28\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{70\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{9}}{9\,n}}+7\,{\frac{{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{8}}{n}}+4\,{\frac{{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{4\,b{a}^{7} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{5}}{5\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+5*n)*(a+b*x^n)^8,x)

[Out]

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

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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+5*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33096, size = 288, normalized size = 2.72 \begin{align*} \frac{495 \, b^{8} x^{13 \, n} + 4290 \, a b^{7} x^{12 \, n} + 16380 \, a^{2} b^{6} x^{11 \, n} + 36036 \, a^{3} b^{5} x^{10 \, n} + 50050 \, a^{4} b^{4} x^{9 \, n} + 45045 \, a^{5} b^{3} x^{8 \, n} + 25740 \, a^{6} b^{2} x^{7 \, n} + 8580 \, a^{7} b x^{6 \, n} + 1287 \, a^{8} x^{5 \, n}}{6435 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+5*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/6435*(495*b^8*x^(13*n) + 4290*a*b^7*x^(12*n) + 16380*a^2*b^6*x^(11*n) + 36036*a^3*b^5*x^(10*n) + 50050*a^4*b
^4*x^(9*n) + 45045*a^5*b^3*x^(8*n) + 25740*a^6*b^2*x^(7*n) + 8580*a^7*b*x^(6*n) + 1287*a^8*x^(5*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+5*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^{5 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+5*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(5*n - 1), x)