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]

Log[2 + b*x^n]/(b*n)

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Rubi [A]  time = 0.0049866, antiderivative size = 15, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0021038, size = 15, normalized size = 1. \[ \frac{\log \left (b x^n+2\right )}{b n} \]

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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Maple [A]  time = 0.012, size = 18, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.940252, size = 20, normalized size = 1.33 \begin{align*} \frac{\log \left (b x^{n} + 2\right )}{b n} \end{align*}

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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Fricas [A]  time = 1.02488, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (b x^{n} + 2\right )}{b n} \end{align*}

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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Sympy [A]  time = 2.29555, size = 27, normalized size = 1.8 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{2} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{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} \end{align*}

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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Giac [A]  time = 1.11453, size = 22, normalized size = 1.47 \begin{align*} \frac{\log \left ({\left | b x^{n} + 2 \right |}\right )}{b n} \end{align*}

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)