3.2578 \(\int x^{-1-3 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=133 \[ -\frac{28 a^6 b^2 x^{-n}}{n}+\frac{70 a^4 b^4 x^n}{n}+\frac{28 a^3 b^5 x^{2 n}}{n}+\frac{28 a^2 b^6 x^{3 n}}{3 n}+56 a^5 b^3 \log (x)-\frac{4 a^7 b x^{-2 n}}{n}-\frac{a^8 x^{-3 n}}{3 n}+\frac{2 a b^7 x^{4 n}}{n}+\frac{b^8 x^{5 n}}{5 n} \]

[Out]

-a^8/(3*n*x^(3*n)) - (4*a^7*b)/(n*x^(2*n)) - (28*a^6*b^2)/(n*x^n) + (70*a^4*b^4*x^n)/n + (28*a^3*b^5*x^(2*n))/
n + (28*a^2*b^6*x^(3*n))/(3*n) + (2*a*b^7*x^(4*n))/n + (b^8*x^(5*n))/(5*n) + 56*a^5*b^3*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0584728, antiderivative size = 133, 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} \[ -\frac{28 a^6 b^2 x^{-n}}{n}+\frac{70 a^4 b^4 x^n}{n}+\frac{28 a^3 b^5 x^{2 n}}{n}+\frac{28 a^2 b^6 x^{3 n}}{3 n}+56 a^5 b^3 \log (x)-\frac{4 a^7 b x^{-2 n}}{n}-\frac{a^8 x^{-3 n}}{3 n}+\frac{2 a b^7 x^{4 n}}{n}+\frac{b^8 x^{5 n}}{5 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 3*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(3*n*x^(3*n)) - (4*a^7*b)/(n*x^(2*n)) - (28*a^6*b^2)/(n*x^n) + (70*a^4*b^4*x^n)/n + (28*a^3*b^5*x^(2*n))/
n + (28*a^2*b^6*x^(3*n))/(3*n) + (2*a*b^7*x^(4*n))/n + (b^8*x^(5*n))/(5*n) + 56*a^5*b^3*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^{-1-3 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^4} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (70 a^4 b^4+\frac{a^8}{x^4}+\frac{8 a^7 b}{x^3}+\frac{28 a^6 b^2}{x^2}+\frac{56 a^5 b^3}{x}+56 a^3 b^5 x+28 a^2 b^6 x^2+8 a b^7 x^3+b^8 x^4\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-3 n}}{3 n}-\frac{4 a^7 b x^{-2 n}}{n}-\frac{28 a^6 b^2 x^{-n}}{n}+\frac{70 a^4 b^4 x^n}{n}+\frac{28 a^3 b^5 x^{2 n}}{n}+\frac{28 a^2 b^6 x^{3 n}}{3 n}+\frac{2 a b^7 x^{4 n}}{n}+\frac{b^8 x^{5 n}}{5 n}+56 a^5 b^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0607483, size = 114, normalized size = 0.86 \[ \frac{-28 a^6 b^2 x^{-n}+70 a^4 b^4 x^n+28 a^3 b^5 x^{2 n}+\frac{28}{3} a^2 b^6 x^{3 n}+56 a^5 b^3 n \log (x)-4 a^7 b x^{-2 n}-\frac{1}{3} a^8 x^{-3 n}+2 a b^7 x^{4 n}+\frac{1}{5} b^8 x^{5 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 3*n)*(a + b*x^n)^8,x]

[Out]

(-a^8/(3*x^(3*n)) - (4*a^7*b)/x^(2*n) - (28*a^6*b^2)/x^n + 70*a^4*b^4*x^n + 28*a^3*b^5*x^(2*n) + (28*a^2*b^6*x
^(3*n))/3 + 2*a*b^7*x^(4*n) + (b^8*x^(5*n))/5 + 56*a^5*b^3*n*Log[x])/n

________________________________________________________________________________________

Maple [A]  time = 0.026, size = 128, normalized size = 1. \begin{align*} 56\,{a}^{5}{b}^{3}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{5}}{5\,n}}+2\,{\frac{{b}^{7}a \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{28\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{3}}{3\,n}}+28\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{2}}{n}}+70\,{\frac{{x}^{n}{a}^{4}{b}^{4}}{n}}-28\,{\frac{{a}^{6}{b}^{2}}{n{x}^{n}}}-4\,{\frac{b{a}^{7}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{{a}^{8}}{3\,n \left ({x}^{n} \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-3*n)*(a+b*x^n)^8,x)

[Out]

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

________________________________________________________________________________________

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^(-1-3*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.36124, size = 265, normalized size = 1.99 \begin{align*} \frac{840 \, a^{5} b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 30 \, a b^{7} x^{7 \, n} + 140 \, a^{2} b^{6} x^{6 \, n} + 420 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{6} b^{2} x^{2 \, n} - 60 \, a^{7} b x^{n} - 5 \, a^{8}}{15 \, n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/15*(840*a^5*b^3*n*x^(3*n)*log(x) + 3*b^8*x^(8*n) + 30*a*b^7*x^(7*n) + 140*a^2*b^6*x^(6*n) + 420*a^3*b^5*x^(5
*n) + 1050*a^4*b^4*x^(4*n) - 420*a^6*b^2*x^(2*n) - 60*a^7*b*x^n - 5*a^8)/(n*x^(3*n))

________________________________________________________________________________________

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**(-1-3*n)*(a+b*x**n)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.26142, size = 157, normalized size = 1.18 \begin{align*} \frac{840 \, a^{5} b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 30 \, a b^{7} x^{7 \, n} + 140 \, a^{2} b^{6} x^{6 \, n} + 420 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{6} b^{2} x^{2 \, n} - 60 \, a^{7} b x^{n} - 5 \, a^{8}}{15 \, n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

1/15*(840*a^5*b^3*n*x^(3*n)*log(x) + 3*b^8*x^(8*n) + 30*a*b^7*x^(7*n) + 140*a^2*b^6*x^(6*n) + 420*a^3*b^5*x^(5
*n) + 1050*a^4*b^4*x^(4*n) - 420*a^6*b^2*x^(2*n) - 60*a^7*b*x^n - 5*a^8)/(n*x^(3*n))