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]

-1/5*a/n/(x^(5*n))-1/4*b/n/(x^(4*n))

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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]

-1/20*(4*a + 5*b*x^n)/(n*x^(5*n))

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fricas [A]  time = 0.76, size = 22, normalized size = 0.81 \[ -\frac {5 \, b x^{n} + 4 \, a}{20 \, n x^{5 \, n}} \]

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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giac [A]  time = 0.16, size = 22, normalized size = 0.81 \[ -\frac {5 \, b x^{n} + 4 \, a}{20 \, n x^{5 \, n}} \]

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))

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maple [A]  time = 0.02, size = 27, normalized size = 1.00 \[ \left (-\frac {b \,{\mathrm e}^{n \ln \relax (x )}}{4 n}-\frac {a}{5 n}\right ) {\mathrm e}^{-5 n \ln \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/5*a/n-1/4*b/n*exp(n*ln(x)))/exp(n*ln(x))^5

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maxima [A]  time = 0.54, size = 27, normalized size = 1.00 \[ -\frac {a}{5 \, n x^{5 \, n}} - \frac {b}{4 \, n x^{4 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*a/(n*x^(5*n)) - 1/4*b/(n*x^(4*n))

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mupad [B]  time = 1.22, size = 22, normalized size = 0.81 \[ -\frac {4\,a+5\,b\,x^n}{20\,n\,x^{5\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)/x^(5*n + 1),x)

[Out]

-(4*a + 5*b*x^n)/(20*n*x^(5*n))

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sympy [A]  time = 5.04, size = 27, normalized size = 1.00 \[ \begin {cases} - \frac {a x^{- 5 n}}{5 n} - \frac {b x^{- 4 n}}{4 n} & \text {for}\: n \neq 0 \\\left (a + b\right ) \log {\relax (x )} & \text {otherwise} \end {cases} \]

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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