Optimal. Leaf size=15 \[ \frac {\log \left (a x^n+b\right )}{a n} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1593, 260} \[ \frac {\log \left (a x^n+b\right )}{a n} \]
Antiderivative was successfully verified.
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Rule 260
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{a x+b x^{1-n}} \, dx &=\int \frac {x^{-1+n}}{b+a x^n} \, dx\\ &=\frac {\log \left (b+a x^n\right )}{a n}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\log \left (a x^n+b\right )}{a n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 28, normalized size = 1.87 \[ \frac {{\left (n - 1\right )} \log \relax (x) + \log \left (a x + b x^{-n + 1}\right )}{a n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a x + b x^{-n + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 41, normalized size = 2.73 \[ \frac {\ln \relax (x )}{a}-\frac {\ln \relax (x )}{a n}+\frac {\ln \left (a x +b \,{\mathrm e}^{\left (-n +1\right ) \ln \relax (x )}\right )}{a n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 19, normalized size = 1.27 \[ \frac {\log \left (\frac {a x^{n} + b}{a}\right )}{a n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.22, size = 34, normalized size = 2.27 \[ \frac {\ln \left (a\,x+b\,x^{1-n}\right )}{a\,n}+\frac {\ln \relax (x)\,\left (n-1\right )}{a\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.14, size = 39, normalized size = 2.60 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{a + b} & \text {for}\: n = 0 \\\frac {x^{n}}{b n} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{a} + \frac {\log {\left (\frac {a}{b} + x^{- n} \right )}}{a n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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