Optimal. Leaf size=23 \[ \frac {\log (x)}{a}-\frac {\log \left (a+b x^n\right )}{a n} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1593, 266, 36, 29, 31} \[ \frac {\log (x)}{a}-\frac {\log \left (a+b x^n\right )}{a n} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{a x+b x^{1+n}} \, dx &=\int \frac {1}{x \left (a+b x^n\right )} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{a n}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^n\right )}{a n}\\ &=\frac {\log (x)}{a}-\frac {\log \left (a+b x^n\right )}{a n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.96 \[ \frac {n \log (x)-\log \left (a+b x^n\right )}{a n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 28, normalized size = 1.22 \[ \frac {{\left (n + 1\right )} \log \relax (x) - \log \left (a x + b x^{n + 1}\right )}{a n} \]
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 [A] time = 0.05, size = 39, normalized size = 1.70 \[ \frac {\ln \relax (x )}{a}+\frac {\ln \relax (x )}{a n}-\frac {\ln \left (a x +b \,{\mathrm e}^{\left (n +1\right ) \ln \relax (x )}\right )}{a n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 27, normalized size = 1.17 \[ \frac {\log \relax (x)}{a} - \frac {\log \left (\frac {b x^{n} + a}{b}\right )}{a n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.23, size = 31, normalized size = 1.35 \[ \frac {\ln \relax (x)\,\left (n+1\right )}{a\,n}-\frac {\ln \left (x\,\left (a+b\,x^n\right )\right )}{a\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.81, size = 41, normalized size = 1.78 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{a + b} & \text {for}\: n = 0 \\- \frac {x^{- n}}{b n} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{n} \right )}}{a n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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