3.2612 \(\int \frac{x^{-1+n}}{2+b x^n} \, dx\)

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

[Out]

Log[2 + b*x^n]/(b*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.0187385, 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 \[ \frac{\log \left (b x^n+2\right )}{b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n)/(2 + b*x^n),x]

[Out]

Log[2 + b*x^n]/(b*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.41564, size = 10, normalized size = 0.67 \[ \frac{\log{\left (b x^{n} + 2 \right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+n)/(2+b*x**n),x)

[Out]

log(b*x**n + 2)/(b*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00394347, 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)

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 18, normalized size = 1.2 \[{\frac{\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{bn}} \]

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)))

_______________________________________________________________________________________

Maxima [A]  time = 1.43609, size = 20, normalized size = 1.33 \[ \frac{\log \left (b x^{n} + 2\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x^n + 2)/(b*n)

_______________________________________________________________________________________

Fricas [A]  time = 0.217868, size = 20, normalized size = 1.33 \[ \frac{\log \left (b x^{n} + 2\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x^n + 2)/(b*n)

_______________________________________________________________________________________

Sympy [A]  time = 8.26104, size = 27, normalized size = 1.8 \[ \begin{cases} \frac{\log{\left (x \right )}}{2} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{b + 2} & \text{for}\: n = 0 \\\frac{x^{n}}{2 n} & \text{for}\: b = 0 \\\frac{\log{\left (x^{n} + \frac{2}{b} \right )}}{b n} & \text{otherwise} \end{cases} \]

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))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219021, size = 22, normalized size = 1.47 \[ \frac{{\rm ln}\left ({\left | b x^{n} + 2 \right |}\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(b*x^n + 2))/(b*n)