3.2559 \(\int x^{-1-4 n} (a+b x^n)^5 \, dx\)

Optimal. Leaf size=82 \[ -\frac{5 a^3 b^2 x^{-2 n}}{n}-\frac{10 a^2 b^3 x^{-n}}{n}-\frac{5 a^4 b x^{-3 n}}{3 n}-\frac{a^5 x^{-4 n}}{4 n}+5 a b^4 \log (x)+\frac{b^5 x^n}{n} \]

[Out]

-a^5/(4*n*x^(4*n)) - (5*a^4*b)/(3*n*x^(3*n)) - (5*a^3*b^2)/(n*x^(2*n)) - (10*a^2*b^3)/(n*x^n) + (b^5*x^n)/n +
5*a*b^4*Log[x]

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Rubi [A]  time = 0.0350119, antiderivative size = 82, 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} \[ -\frac{5 a^3 b^2 x^{-2 n}}{n}-\frac{10 a^2 b^3 x^{-n}}{n}-\frac{5 a^4 b x^{-3 n}}{3 n}-\frac{a^5 x^{-4 n}}{4 n}+5 a b^4 \log (x)+\frac{b^5 x^n}{n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 4*n)*(a + b*x^n)^5,x]

[Out]

-a^5/(4*n*x^(4*n)) - (5*a^4*b)/(3*n*x^(3*n)) - (5*a^3*b^2)/(n*x^(2*n)) - (10*a^2*b^3)/(n*x^n) + (b^5*x^n)/n +
5*a*b^4*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^{-1-4 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^5} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^5+\frac{a^5}{x^5}+\frac{5 a^4 b}{x^4}+\frac{10 a^3 b^2}{x^3}+\frac{10 a^2 b^3}{x^2}+\frac{5 a b^4}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-4 n}}{4 n}-\frac{5 a^4 b x^{-3 n}}{3 n}-\frac{5 a^3 b^2 x^{-2 n}}{n}-\frac{10 a^2 b^3 x^{-n}}{n}+\frac{b^5 x^n}{n}+5 a b^4 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0407851, size = 72, normalized size = 0.88 \[ \frac{-5 a^3 b^2 x^{-2 n}-10 a^2 b^3 x^{-n}-\frac{5}{3} a^4 b x^{-3 n}-\frac{1}{4} a^5 x^{-4 n}+5 a b^4 n \log (x)+b^5 x^n}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 4*n)*(a + b*x^n)^5,x]

[Out]

(-a^5/(4*x^(4*n)) - (5*a^4*b)/(3*x^(3*n)) - (5*a^3*b^2)/x^(2*n) - (10*a^2*b^3)/x^n + b^5*x^n + 5*a*b^4*n*Log[x
])/n

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Maple [A]  time = 0.016, size = 97, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ({\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{n}}+5\,a{b}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}-{\frac{{a}^{5}}{4\,n}}-10\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-5\,{\frac{{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{5\,{a}^{4}b{{\rm e}^{n\ln \left ( x \right ) }}}{3\,n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-4*n)*(a+b*x^n)^5,x)

[Out]

(b^5/n*exp(n*ln(x))^5+5*a*b^4*ln(x)*exp(n*ln(x))^4-1/4*a^5/n-10*a^2*b^3/n*exp(n*ln(x))^3-5*a^3*b^2/n*exp(n*ln(
x))^2-5/3*a^4*b/n*exp(n*ln(x)))/exp(n*ln(x))^4

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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^(-1-4*n)*(a+b*x^n)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29766, size = 174, normalized size = 2.12 \begin{align*} \frac{60 \, a b^{4} n x^{4 \, n} \log \left (x\right ) + 12 \, b^{5} x^{5 \, n} - 120 \, a^{2} b^{3} x^{3 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 20 \, a^{4} b x^{n} - 3 \, a^{5}}{12 \, n x^{4 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^5,x, algorithm="fricas")

[Out]

1/12*(60*a*b^4*n*x^(4*n)*log(x) + 12*b^5*x^(5*n) - 120*a^2*b^3*x^(3*n) - 60*a^3*b^2*x^(2*n) - 20*a^4*b*x^n - 3
*a^5)/(n*x^(4*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**(-1-4*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.2369, size = 104, normalized size = 1.27 \begin{align*} \frac{60 \, a b^{4} n x^{4 \, n} \log \left (x\right ) + 12 \, b^{5} x^{5 \, n} - 120 \, a^{2} b^{3} x^{3 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 20 \, a^{4} b x^{n} - 3 \, a^{5}}{12 \, n x^{4 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^5,x, algorithm="giac")

[Out]

1/12*(60*a*b^4*n*x^(4*n)*log(x) + 12*b^5*x^(5*n) - 120*a^2*b^3*x^(3*n) - 60*a^3*b^2*x^(2*n) - 20*a^4*b*x^n - 3
*a^5)/(n*x^(4*n))