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