3.2560 \(\int x^{-1-5 n} (a+b x^n)^5 \, dx\)

Optimal. Leaf size=86 \[ -\frac{10 a^3 b^2 x^{-3 n}}{3 n}-\frac{5 a^2 b^3 x^{-2 n}}{n}-\frac{5 a^4 b x^{-4 n}}{4 n}-\frac{a^5 x^{-5 n}}{5 n}-\frac{5 a b^4 x^{-n}}{n}+b^5 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0372315, antiderivative size = 86, 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} \[ -\frac{10 a^3 b^2 x^{-3 n}}{3 n}-\frac{5 a^2 b^3 x^{-2 n}}{n}-\frac{5 a^4 b x^{-4 n}}{4 n}-\frac{a^5 x^{-5 n}}{5 n}-\frac{5 a b^4 x^{-n}}{n}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 5*n)*(a + b*x^n)^5,x]

[Out]

-a^5/(5*n*x^(5*n)) - (5*a^4*b)/(4*n*x^(4*n)) - (10*a^3*b^2)/(3*n*x^(3*n)) - (5*a^2*b^3)/(n*x^(2*n)) - (5*a*b^4
)/(n*x^n) + b^5*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^{-1-5 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^6} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^5}{x^6}+\frac{5 a^4 b}{x^5}+\frac{10 a^3 b^2}{x^4}+\frac{10 a^2 b^3}{x^3}+\frac{5 a b^4}{x^2}+\frac{b^5}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-5 n}}{5 n}-\frac{5 a^4 b x^{-4 n}}{4 n}-\frac{10 a^3 b^2 x^{-3 n}}{3 n}-\frac{5 a^2 b^3 x^{-2 n}}{n}-\frac{5 a b^4 x^{-n}}{n}+b^5 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0604278, size = 69, normalized size = 0.8 \[ b^5 \log (x)-\frac{a x^{-5 n} \left (200 a^2 b^2 x^{2 n}+75 a^3 b x^n+12 a^4+300 a b^3 x^{3 n}+300 b^4 x^{4 n}\right )}{60 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(12*a^4 + 75*a^3*b*x^n + 200*a^2*b^2*x^(2*n) + 300*a*b^3*x^(3*n) + 300*b^4*x^(4*n)))/(60*n*x^(5*n)) + b^5*
Log[x]

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Maple [A]  time = 0.015, size = 97, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}} \left ({b}^{5}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}-{\frac{{a}^{5}}{5\,n}}-5\,{\frac{a{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{n}}-5\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-{\frac{10\,{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{3\,n}}-{\frac{5\,{a}^{4}b{{\rm e}^{n\ln \left ( x \right ) }}}{4\,n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^5*ln(x)*exp(n*ln(x))^5-1/5*a^5/n-5*a*b^4/n*exp(n*ln(x))^4-5*a^2*b^3/n*exp(n*ln(x))^3-10/3*a^3*b^2/n*exp(n*l
n(x))^2-5/4*a^4*b/n*exp(n*ln(x)))/exp(n*ln(x))^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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33071, size = 178, normalized size = 2.07 \begin{align*} \frac{60 \, b^{5} n x^{5 \, n} \log \left (x\right ) - 300 \, a b^{4} x^{4 \, n} - 300 \, a^{2} b^{3} x^{3 \, n} - 200 \, a^{3} b^{2} x^{2 \, n} - 75 \, a^{4} b x^{n} - 12 \, a^{5}}{60 \, n x^{5 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*b^5*n*x^(5*n)*log(x) - 300*a*b^4*x^(4*n) - 300*a^2*b^3*x^(3*n) - 200*a^3*b^2*x^(2*n) - 75*a^4*b*x^n -
12*a^5)/(n*x^(5*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)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.20144, size = 104, normalized size = 1.21 \begin{align*} \frac{60 \, b^{5} n x^{5 \, n} \log \left (x\right ) - 300 \, a b^{4} x^{4 \, n} - 300 \, a^{2} b^{3} x^{3 \, n} - 200 \, a^{3} b^{2} x^{2 \, n} - 75 \, a^{4} b x^{n} - 12 \, a^{5}}{60 \, n x^{5 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*b^5*n*x^(5*n)*log(x) - 300*a*b^4*x^(4*n) - 300*a^2*b^3*x^(3*n) - 200*a^3*b^2*x^(2*n) - 75*a^4*b*x^n -
12*a^5)/(n*x^(5*n))