3.2635 \(\int \frac{x^{-1-n}}{a+b x^n} \, dx\)

Optimal. Leaf size=38 \[ \frac{b \log \left (a+b x^n\right )}{a^2 n}-\frac{b \log (x)}{a^2}-\frac{x^{-n}}{a n} \]

[Out]

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

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Rubi [A]  time = 0.0646212, antiderivative size = 38, 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 \[ \frac{b \log \left (a+b x^n\right )}{a^2 n}-\frac{b \log (x)}{a^2}-\frac{x^{-n}}{a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n)/(a + b*x^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.76393, size = 34, normalized size = 0.89 \[ - \frac{x^{- n}}{a n} - \frac{b \log{\left (x^{n} \right )}}{a^{2} n} + \frac{b \log{\left (a + b x^{n} \right )}}{a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0235453, size = 32, normalized size = 0.84 \[ \frac{b \log \left (a x^{-n}+b\right )}{a^2 n}-\frac{x^{-n}}{a n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0., size = 50, normalized size = 1.3 \[{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( -{\frac{1}{an}}-{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}}} \right ) }+{\frac{b\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{2}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-n)/(a+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.44099, size = 57, normalized size = 1.5 \[ -\frac{b \log \left (x\right )}{a^{2}} - \frac{x^{-n}}{a n} + \frac{b \log \left (\frac{b x^{n} + a}{b}\right )}{a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227185, size = 50, normalized size = 1.32 \[ -\frac{b n x^{n} \log \left (x\right ) - b x^{n} \log \left (b x^{n} + a\right ) + a}{a^{2} n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*n*x^n*log(x) - b*x^n*log(b*x^n + a) + a)/(a^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)/(a+b*x**n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(-n - 1)/(b*x^n + a), x)