3.2527 \(\int x^{-1-5 n} (a+b x^n) \, dx\)

Optimal. Leaf size=27 \[ -\frac{a x^{-5 n}}{5 n}-\frac{b x^{-4 n}}{4 n} \]

[Out]

-a/(5*n*x^(5*n)) - b/(4*n*x^(4*n))

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Rubi [A]  time = 0.0077876, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {14} \[ -\frac{a x^{-5 n}}{5 n}-\frac{b x^{-4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 5*n)*(a + b*x^n),x]

[Out]

-a/(5*n*x^(5*n)) - b/(4*n*x^(4*n))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x^{-1-5 n} \left (a+b x^n\right ) \, dx &=\int \left (a x^{-1-5 n}+b x^{-1-4 n}\right ) \, dx\\ &=-\frac{a x^{-5 n}}{5 n}-\frac{b x^{-4 n}}{4 n}\\ \end{align*}

Mathematica [A]  time = 0.0102061, size = 22, normalized size = 0.81 \[ -\frac{x^{-5 n} \left (4 a+5 b x^n\right )}{20 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 5*n)*(a + b*x^n),x]

[Out]

-(4*a + 5*b*x^n)/(20*n*x^(5*n))

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Maple [A]  time = 0.016, size = 27, normalized size = 1. \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}} \left ( -{\frac{a}{5\,n}}-{\frac{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),x)

[Out]

(-1/5*a/n-1/4*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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.997749, size = 47, normalized size = 1.74 \begin{align*} -\frac{5 \, b x^{n} + 4 \, a}{20 \, 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),x, algorithm="fricas")

[Out]

-1/20*(5*b*x^n + 4*a)/(n*x^(5*n))

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Sympy [A]  time = 4.65464, size = 27, normalized size = 1. \begin{align*} \begin{cases} - \frac{a x^{- 5 n}}{5 n} - \frac{b x^{- 4 n}}{4 n} & \text{for}\: n \neq 0 \\\left (a + b\right ) \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*x**(-5*n)/(5*n) - b*x**(-4*n)/(4*n), Ne(n, 0)), ((a + b)*log(x), True))

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Giac [A]  time = 1.17589, size = 30, normalized size = 1.11 \begin{align*} -\frac{5 \, b x^{n} + 4 \, a}{20 \, 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),x, algorithm="giac")

[Out]

-1/20*(5*b*x^n + 4*a)/(n*x^(5*n))