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

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

[Out]

Log[b + a*x^n]/(a*n)

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Rubi [A]  time = 0.0200165, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\log \left (a x^n+b\right )}{a n} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^(1 - n))^(-1),x]

[Out]

Log[b + a*x^n]/(a*n)

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Rubi in Sympy [A]  time = 2.99698, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a x^{n} + b \right )}}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a*x**n + b)/(a*n)

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Mathematica [A]  time = 0.00693275, size = 15, normalized size = 1. \[ \frac{\log \left (a x^n+b\right )}{a n} \]

Antiderivative was successfully verified.

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

[Out]

Log[b + a*x^n]/(a*n)

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Maple [B]  time = 0.018, size = 41, normalized size = 2.7 \[ -{\frac{\ln \left ( x \right ) }{an}}+{\frac{\ln \left ( x \right ) }{a}}+{\frac{\ln \left ( ax+b{{\rm e}^{ \left ( 1-n \right ) \ln \left ( x \right ) }} \right ) }{an}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x+b*x^(1-n)),x)

[Out]

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

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Maxima [A]  time = 1.38654, size = 26, normalized size = 1.73 \[ \frac{\log \left (\frac{a x^{n} + b}{a}\right )}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log((a*x^n + b)/a)/(a*n)

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Fricas [A]  time = 0.237165, size = 38, normalized size = 2.53 \[ \frac{{\left (n - 1\right )} \log \left (x\right ) + \log \left (a x + b x^{-n + 1}\right )}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.52288, size = 39, normalized size = 2.6 \[ \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{x^{n}}{b n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a} + \frac{\log{\left (\frac{a}{b} + x^{- n} \right )}}{a n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x+b*x**(1-n)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(a*x + b*x^(-n + 1)), x)