3.2594 \(\int \frac{x^{-1+n}}{a+b x^n} \, dx\)

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

[Out]

Log[a + b*x^n]/(b*n)

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Rubi [A]  time = 0.0198588, 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 \[ \frac{\log \left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n)/(a + b*x^n),x]

[Out]

Log[a + b*x^n]/(b*n)

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Rubi in Sympy [A]  time = 2.46656, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a + b x^{n} \right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x**n)/(b*n)

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Mathematica [A]  time = 0.00406986, size = 15, normalized size = 1. \[ \frac{\log \left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x^n]/(b*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)/(a+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.43817, size = 20, normalized size = 1.33 \[ \frac{\log \left (b x^{n} + a\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x^n + a)/(b*n)

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Fricas [A]  time = 0.218754, size = 20, normalized size = 1.33 \[ \frac{\log \left (b x^{n} + a\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x^n + a)/(b*n)

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Sympy [A]  time = 8.18411, size = 27, normalized size = 1.8 \[ \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{x^{n}}{a n} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + x^{n} \right )}}{b n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((log(x)/a, Eq(b, 0) & Eq(n, 0)), (log(x)/(a + b), Eq(n, 0)), (x**n/(a*
n), Eq(b, 0)), (log(a/b + x**n)/(b*n), True))

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GIAC/XCAS [A]  time = 0.215302, size = 22, normalized size = 1.47 \[ \frac{{\rm ln}\left ({\left | b x^{n} + a \right |}\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(b*x^n + a))/(b*n)