3.2614 \(\int \frac{x^{-1-n}}{2+b x^n} \, dx\)

Optimal. Leaf size=36 \[ \frac{b \log \left (b x^n+2\right )}{4 n}-\frac{1}{4} b \log (x)-\frac{x^{-n}}{2 n} \]

[Out]

-1/(2*n*x^n) - (b*Log[x])/4 + (b*Log[2 + b*x^n])/(4*n)

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Rubi [A]  time = 0.0506997, antiderivative size = 36, 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 \[ \frac{b \log \left (b x^n+2\right )}{4 n}-\frac{1}{4} b \log (x)-\frac{x^{-n}}{2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n)/(2 + b*x^n),x]

[Out]

-1/(2*n*x^n) - (b*Log[x])/4 + (b*Log[2 + b*x^n])/(4*n)

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Rubi in Sympy [A]  time = 7.52773, size = 31, normalized size = 0.86 \[ - \frac{b \log{\left (x^{n} \right )}}{4 n} + \frac{b \log{\left (b x^{n} + 2 \right )}}{4 n} - \frac{x^{- n}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*log(x**n)/(4*n) + b*log(b*x**n + 2)/(4*n) - x**(-n)/(2*n)

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Mathematica [A]  time = 0.0186937, size = 31, normalized size = 0.86 \[ \frac{b \log \left (b+2 x^{-n}\right )}{4 n}-\frac{x^{-n}}{2 n} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*n*x^n) + (b*Log[b + 2/x^n])/(4*n)

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Maple [A]  time = 0.026, size = 42, normalized size = 1.2 \[{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( -{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{4}}-{\frac{1}{2\,n}} \right ) }+{\frac{b\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{4\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-n)/(2+b*x^n),x)

[Out]

(-1/4*b*ln(x)*exp(n*ln(x))-1/2/n)/exp(n*ln(x))+1/4*b/n*ln(2+b*exp(n*ln(x)))

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Maxima [A]  time = 1.44282, size = 46, normalized size = 1.28 \[ -\frac{1}{4} \, b \log \left (x\right ) + \frac{b \log \left (\frac{b x^{n} + 2}{b}\right )}{4 \, n} - \frac{x^{-n}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*b*log(x) + 1/4*b*log((b*x^n + 2)/b)/n - 1/2*x^(-n)/n

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Fricas [A]  time = 0.229114, size = 46, normalized size = 1.28 \[ -\frac{b n x^{n} \log \left (x\right ) - b x^{n} \log \left (b x^{n} + 2\right ) + 2}{4 \, n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(b*n*x^n*log(x) - b*x^n*log(b*x^n + 2) + 2)/(n*x^n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(-n - 1)/(b*x^n + 2), x)