3.2561 \(\int x^{-1-6 n} (a+b x^n)^5 \, dx\)

Optimal. Leaf size=24 \[ -\frac{x^{-6 n} \left (a+b x^n\right )^6}{6 a n} \]

[Out]

-(a + b*x^n)^6/(6*a*n*x^(6*n))

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Rubi [A]  time = 0.004856, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ -\frac{x^{-6 n} \left (a+b x^n\right )^6}{6 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 6*n)*(a + b*x^n)^5,x]

[Out]

-(a + b*x^n)^6/(6*a*n*x^(6*n))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^{-1-6 n} \left (a+b x^n\right )^5 \, dx &=-\frac{x^{-6 n} \left (a+b x^n\right )^6}{6 a n}\\ \end{align*}

Mathematica [A]  time = 0.008275, size = 24, normalized size = 1. \[ -\frac{x^{-6 n} \left (a+b x^n\right )^6}{6 a n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 6*n)*(a + b*x^n)^5,x]

[Out]

-(a + b*x^n)^6/(6*a*n*x^(6*n))

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Maple [B]  time = 0.02, size = 88, normalized size = 3.7 \begin{align*} -{\frac{{b}^{5}}{n{x}^{n}}}-{\frac{5\,a{b}^{4}}{2\,n \left ({x}^{n} \right ) ^{2}}}-{\frac{10\,{a}^{2}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{5\,{a}^{3}{b}^{2}}{2\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{{a}^{4}b}{n \left ({x}^{n} \right ) ^{5}}}-{\frac{{a}^{5}}{6\,n \left ({x}^{n} \right ) ^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-6*n)*(a+b*x^n)^5,x)

[Out]

-b^5/n/(x^n)-5/2*a*b^4/n/(x^n)^2-10/3*a^2*b^3/n/(x^n)^3-5/2*a^3*b^2/n/(x^n)^4-a^4*b/n/(x^n)^5-1/6*a^5/n/(x^n)^
6

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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^(-1-6*n)*(a+b*x^n)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.33857, size = 155, normalized size = 6.46 \begin{align*} -\frac{6 \, b^{5} x^{5 \, n} + 15 \, a b^{4} x^{4 \, n} + 20 \, a^{2} b^{3} x^{3 \, n} + 15 \, a^{3} b^{2} x^{2 \, n} + 6 \, a^{4} b x^{n} + a^{5}}{6 \, n x^{6 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-6*n)*(a+b*x^n)^5,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*x^(5*n) + 15*a*b^4*x^(4*n) + 20*a^2*b^3*x^(3*n) + 15*a^3*b^2*x^(2*n) + 6*a^4*b*x^n + a^5)/(n*x^(6*
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**(-1-6*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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Giac [B]  time = 1.21082, size = 97, normalized size = 4.04 \begin{align*} -\frac{6 \, b^{5} x^{5 \, n} + 15 \, a b^{4} x^{4 \, n} + 20 \, a^{2} b^{3} x^{3 \, n} + 15 \, a^{3} b^{2} x^{2 \, n} + 6 \, a^{4} b x^{n} + a^{5}}{6 \, n x^{6 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-6*n)*(a+b*x^n)^5,x, algorithm="giac")

[Out]

-1/6*(6*b^5*x^(5*n) + 15*a*b^4*x^(4*n) + 20*a^2*b^3*x^(3*n) + 15*a^3*b^2*x^(2*n) + 6*a^4*b*x^n + a^5)/(n*x^(6*
n))