3.2619 \(\int \frac {x^{-1+n}}{2+b x^n} \, dx\)

Optimal. Leaf size=15 \[ \frac {\log \left (b x^n+2\right )}{b n} \]

[Out]

ln(2+b*x^n)/b/n

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {260} \[ \frac {\log \left (b x^n+2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)/(2 + b*x^n),x]

[Out]

Log[2 + b*x^n]/(b*n)

Rule 260

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

Rubi steps

\begin {align*} \int \frac {x^{-1+n}}{2+b x^n} \, dx &=\frac {\log \left (2+b x^n\right )}{b n}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\log \left (b x^n+2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)/(2 + b*x^n),x]

[Out]

Log[2 + b*x^n]/(b*n)

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fricas [A]  time = 0.53, size = 15, normalized size = 1.00 \[ \frac {\log \left (b x^{n} + 2\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)/(2+b*x^n),x, algorithm="fricas")

[Out]

log(b*x^n + 2)/(b*n)

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giac [A]  time = 0.16, size = 16, normalized size = 1.07 \[ \frac {\log \left ({\left | b x^{n} + 2 \right |}\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)/(2+b*x^n),x, algorithm="giac")

[Out]

log(abs(b*x^n + 2))/(b*n)

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maple [A]  time = 0.02, size = 18, normalized size = 1.20 \[ \frac {\ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+2\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n-1)/(2+b*x^n),x)

[Out]

1/b/n*ln(b*exp(n*ln(x))+2)

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maxima [A]  time = 0.70, size = 15, normalized size = 1.00 \[ \frac {\log \left (b x^{n} + 2\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)/(2+b*x^n),x, algorithm="maxima")

[Out]

log(b*x^n + 2)/(b*n)

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mupad [B]  time = 1.21, size = 15, normalized size = 1.00 \[ \frac {\ln \left (b\,x^n+2\right )}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - 1)/(b*x^n + 2),x)

[Out]

log(b*x^n + 2)/(b*n)

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sympy [A]  time = 2.38, size = 27, normalized size = 1.80 \[ \begin {cases} \frac {\log {\relax (x )}}{2} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{b + 2} & \text {for}\: n = 0 \\\frac {x^{n}}{2 n} & \text {for}\: b = 0 \\\frac {\log {\left (x^{n} + \frac {2}{b} \right )}}{b n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+n)/(2+b*x**n),x)

[Out]

Piecewise((log(x)/2, Eq(b, 0) & Eq(n, 0)), (log(x)/(b + 2), Eq(n, 0)), (x**n/(2*n), Eq(b, 0)), (log(x**n + 2/b
)/(b*n), True))

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