3.2576 \(\int x^{-1-n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=135 \[ \frac{28 a^6 b^2 x^n}{n}+\frac{28 a^5 b^3 x^{2 n}}{n}+\frac{70 a^4 b^4 x^{3 n}}{3 n}+\frac{14 a^3 b^5 x^{4 n}}{n}+\frac{28 a^2 b^6 x^{5 n}}{5 n}+8 a^7 b \log (x)-\frac{a^8 x^{-n}}{n}+\frac{4 a b^7 x^{6 n}}{3 n}+\frac{b^8 x^{7 n}}{7 n} \]

[Out]

-(a^8/(n*x^n)) + (28*a^6*b^2*x^n)/n + (28*a^5*b^3*x^(2*n))/n + (70*a^4*b^4*x^(3*n))/(3*n) + (14*a^3*b^5*x^(4*n
))/n + (28*a^2*b^6*x^(5*n))/(5*n) + (4*a*b^7*x^(6*n))/(3*n) + (b^8*x^(7*n))/(7*n) + 8*a^7*b*Log[x]

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Rubi [A]  time = 0.0573147, antiderivative size = 135, 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} \[ \frac{28 a^6 b^2 x^n}{n}+\frac{28 a^5 b^3 x^{2 n}}{n}+\frac{70 a^4 b^4 x^{3 n}}{3 n}+\frac{14 a^3 b^5 x^{4 n}}{n}+\frac{28 a^2 b^6 x^{5 n}}{5 n}+8 a^7 b \log (x)-\frac{a^8 x^{-n}}{n}+\frac{4 a b^7 x^{6 n}}{3 n}+\frac{b^8 x^{7 n}}{7 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*(a + b*x^n)^8,x]

[Out]

-(a^8/(n*x^n)) + (28*a^6*b^2*x^n)/n + (28*a^5*b^3*x^(2*n))/n + (70*a^4*b^4*x^(3*n))/(3*n) + (14*a^3*b^5*x^(4*n
))/n + (28*a^2*b^6*x^(5*n))/(5*n) + (4*a*b^7*x^(6*n))/(3*n) + (b^8*x^(7*n))/(7*n) + 8*a^7*b*Log[x]

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-n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (28 a^6 b^2+\frac{a^8}{x^2}+\frac{8 a^7 b}{x}+56 a^5 b^3 x+70 a^4 b^4 x^2+56 a^3 b^5 x^3+28 a^2 b^6 x^4+8 a b^7 x^5+b^8 x^6\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-n}}{n}+\frac{28 a^6 b^2 x^n}{n}+\frac{28 a^5 b^3 x^{2 n}}{n}+\frac{70 a^4 b^4 x^{3 n}}{3 n}+\frac{14 a^3 b^5 x^{4 n}}{n}+\frac{28 a^2 b^6 x^{5 n}}{5 n}+\frac{4 a b^7 x^{6 n}}{3 n}+\frac{b^8 x^{7 n}}{7 n}+8 a^7 b \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0594267, size = 116, normalized size = 0.86 \[ \frac{28 a^6 b^2 x^n+28 a^5 b^3 x^{2 n}+\frac{70}{3} a^4 b^4 x^{3 n}+14 a^3 b^5 x^{4 n}+\frac{28}{5} a^2 b^6 x^{5 n}+8 a^7 b n \log (x)-a^8 x^{-n}+\frac{4}{3} a b^7 x^{6 n}+\frac{1}{7} b^8 x^{7 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n)*(a + b*x^n)^8,x]

[Out]

(-(a^8/x^n) + 28*a^6*b^2*x^n + 28*a^5*b^3*x^(2*n) + (70*a^4*b^4*x^(3*n))/3 + 14*a^3*b^5*x^(4*n) + (28*a^2*b^6*
x^(5*n))/5 + (4*a*b^7*x^(6*n))/3 + (b^8*x^(7*n))/7 + 8*a^7*b*n*Log[x])/n

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Maple [A]  time = 0.025, size = 128, normalized size = 1. \begin{align*} 8\,{a}^{7}b\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{4\,{b}^{7}a \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{28\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{5}}{5\,n}}+14\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{70\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{3}}{3\,n}}+28\,{\frac{{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{2}}{n}}+28\,{\frac{{x}^{n}{a}^{6}{b}^{2}}{n}}-{\frac{{a}^{8}}{n{x}^{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*a^7*b*ln(x)+1/7*b^8/n*(x^n)^7+4/3*a*b^7/n*(x^n)^6+28/5*a^2*b^6/n*(x^n)^5+14*a^3*b^5/n*(x^n)^4+70/3*a^4*b^4/n
*(x^n)^3+28*a^5*b^3/n*(x^n)^2+28*a^6*b^2*x^n/n-a^8/n/(x^n)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.46858, size = 271, normalized size = 2.01 \begin{align*} \frac{840 \, a^{7} b n x^{n} \log \left (x\right ) + 15 \, b^{8} x^{8 \, n} + 140 \, a b^{7} x^{7 \, n} + 588 \, a^{2} b^{6} x^{6 \, n} + 1470 \, a^{3} b^{5} x^{5 \, n} + 2450 \, a^{4} b^{4} x^{4 \, n} + 2940 \, a^{5} b^{3} x^{3 \, n} + 2940 \, a^{6} b^{2} x^{2 \, n} - 105 \, a^{8}}{105 \, n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(840*a^7*b*n*x^n*log(x) + 15*b^8*x^(8*n) + 140*a*b^7*x^(7*n) + 588*a^2*b^6*x^(6*n) + 1470*a^3*b^5*x^(5*n
) + 2450*a^4*b^4*x^(4*n) + 2940*a^5*b^3*x^(3*n) + 2940*a^6*b^2*x^(2*n) - 105*a^8)/(n*x^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-n)*(a+b*x**n)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.24473, size = 154, normalized size = 1.14 \begin{align*} \frac{840 \, a^{7} b n x^{n} \log \left (x\right ) + 15 \, b^{8} x^{8 \, n} + 140 \, a b^{7} x^{7 \, n} + 588 \, a^{2} b^{6} x^{6 \, n} + 1470 \, a^{3} b^{5} x^{5 \, n} + 2450 \, a^{4} b^{4} x^{4 \, n} + 2940 \, a^{5} b^{3} x^{3 \, n} + 2940 \, a^{6} b^{2} x^{2 \, n} - 105 \, a^{8}}{105 \, n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(840*a^7*b*n*x^n*log(x) + 15*b^8*x^(8*n) + 140*a*b^7*x^(7*n) + 588*a^2*b^6*x^(6*n) + 1470*a^3*b^5*x^(5*n
) + 2450*a^4*b^4*x^(4*n) + 2940*a^5*b^3*x^(3*n) + 2940*a^6*b^2*x^(2*n) - 105*a^8)/(n*x^n)