Optimal. Leaf size=23 \[ \frac {\log \left (a x^{1-n}+b\right )}{a (1-n)} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1593, 260} \[ \frac {\log \left (a x^{1-n}+b\right )}{a (1-n)} \]
Antiderivative was successfully verified.
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Rule 260
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{a x+b x^n} \, dx &=\int \frac {x^{-n}}{b+a x^{1-n}} \, dx\\ &=\frac {\log \left (b+a x^{1-n}\right )}{a (1-n)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ \frac {\log \left (a x^{1-n}+b\right )}{a (1-n)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 27, normalized size = 1.17 \[ \frac {n \log \relax (x) - \log \left (a x + b x^{n}\right )}{a n - a} \]
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 \[ \int \frac {1}{a x + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 1.57 \[ \frac {n \ln \relax (x )}{\left (n -1\right ) a}-\frac {\ln \left (a x +b \,{\mathrm e}^{n \ln \relax (x )}\right )}{\left (n -1\right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 37, normalized size = 1.61 \[ \frac {n \log \relax (x)}{a {\left (n - 1\right )}} - \frac {\log \left (\frac {a x + b x^{n}}{b}\right )}{a {\left (n - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.26, size = 26, normalized size = 1.13 \[ -\frac {\ln \left (b\,x^n+a\,x\right )-n\,\ln \relax (x)}{a\,\left (n-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 53, normalized size = 2.30 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 1 \\- \frac {x}{b \left (n x^{n} - x^{n}\right )} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{a + b} & \text {for}\: n = 1 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {n \log {\relax (x )}}{a n - a} - \frac {\log {\left (\frac {a x}{b} + x^{n} \right )}}{a n - a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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