3.2611 \(\int \frac{x^{-1+2 n}}{2+b x^n} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^n}{b n}-\frac{2 \log \left (b x^n+2\right )}{b^2 n} \]

[Out]

x^n/(b*n) - (2*Log[2 + b*x^n])/(b^2*n)

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Rubi [A]  time = 0.0444568, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x^n}{b n}-\frac{2 \log \left (b x^n+2\right )}{b^2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 2*n)/(2 + b*x^n),x]

[Out]

x^n/(b*n) - (2*Log[2 + b*x^n])/(b^2*n)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{n}} \frac{1}{b}\, dx}{n} - \frac{2 \log{\left (b x^{n} + 2 \right )}}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0150376, size = 23, normalized size = 0.85 \[ \frac{b x^n-2 \log \left (b x^n+2\right )}{b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

(b*x^n - 2*Log[2 + b*x^n])/(b^2*n)

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Maple [A]  time = 0.026, size = 32, normalized size = 1.2 \[{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{bn}}-2\,{\frac{\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{2}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)/(2+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.44441, size = 42, normalized size = 1.56 \[ \frac{x^{n}}{b n} - \frac{2 \, \log \left (\frac{b x^{n} + 2}{b}\right )}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x^n/(b*n) - 2*log((b*x^n + 2)/b)/(b^2*n)

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Fricas [A]  time = 0.226077, size = 31, normalized size = 1.15 \[ \frac{b x^{n} - 2 \, \log \left (b x^{n} + 2\right )}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x^n - 2*log(b*x^n + 2))/(b^2*n)

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Sympy [A]  time = 45.4758, size = 39, normalized size = 1.44 \[ \begin{cases} \frac{\log{\left (x \right )}}{2} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{b + 2} & \text{for}\: n = 0 \\\frac{x^{2 n}}{4 n} & \text{for}\: b = 0 \\\frac{x^{n}}{b n} - \frac{2 \log{\left (x^{n} + \frac{2}{b} \right )}}{b^{2} n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+2*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**(2*n)
/(4*n), Eq(b, 0)), (x**n/(b*n) - 2*log(x**n + 2/b)/(b**2*n), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{b x^{n} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(2*n - 1)/(b*x^n + 2), x)