3.2573 \(\int x^{-1+2 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=40 \[ \frac{\left (a+b x^n\right )^{10}}{10 b^2 n}-\frac{a \left (a+b x^n\right )^9}{9 b^2 n} \]

[Out]

-(a*(a + b*x^n)^9)/(9*b^2*n) + (a + b*x^n)^10/(10*b^2*n)

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Rubi [A]  time = 0.0185763, antiderivative size = 40, 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} \[ \frac{\left (a+b x^n\right )^{10}}{10 b^2 n}-\frac{a \left (a+b x^n\right )^9}{9 b^2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 2*n)*(a + b*x^n)^8,x]

[Out]

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

Mathematica [A]  time = 0.0266571, size = 27, normalized size = 0.68 \[ -\frac{\left (a-9 b x^n\right ) \left (a+b x^n\right )^9}{90 b^2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 2*n)*(a + b*x^n)^8,x]

[Out]

-((a - 9*b*x^n)*(a + b*x^n)^9)/(90*b^2*n)

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Maple [B]  time = 0.022, size = 136, normalized size = 3.4 \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{10}}{10\,n}}+{\frac{8\,{b}^{7}a \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{7\,{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{8}}{2\,n}}+8\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{35\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{56\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{5}}{5\,n}}+7\,{\frac{{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{8\,b{a}^{7} \left ({x}^{n} \right ) ^{3}}{3\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{2}}{2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+2*n)*(a+b*x^n)^8,x)

[Out]

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

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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+2*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.29129, size = 261, normalized size = 6.52 \begin{align*} \frac{9 \, b^{8} x^{10 \, n} + 80 \, a b^{7} x^{9 \, n} + 315 \, a^{2} b^{6} x^{8 \, n} + 720 \, a^{3} b^{5} x^{7 \, n} + 1050 \, a^{4} b^{4} x^{6 \, n} + 1008 \, a^{5} b^{3} x^{5 \, n} + 630 \, a^{6} b^{2} x^{4 \, n} + 240 \, a^{7} b x^{3 \, n} + 45 \, a^{8} x^{2 \, n}}{90 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/90*(9*b^8*x^(10*n) + 80*a*b^7*x^(9*n) + 315*a^2*b^6*x^(8*n) + 720*a^3*b^5*x^(7*n) + 1050*a^4*b^4*x^(6*n) + 1
008*a^5*b^3*x^(5*n) + 630*a^6*b^2*x^(4*n) + 240*a^7*b*x^(3*n) + 45*a^8*x^(2*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+2*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^{2 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(2*n - 1), x)