Optimal. Leaf size=17 \[ \log \left (a x^{1-n}+b\right )+n \log (x) \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {514, 446, 72} \[ \log \left (a x^{1-n}+b\right )+n \log (x) \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 514
Rubi steps
\begin {align*} \int \frac {x^{-n} \left (a+b n x^{-1+n}\right )}{b+a x^{1-n}} \, dx &=\int \frac {b n+a x^{1-n}}{x \left (b+a x^{1-n}\right )} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {b n+a x}{x (b+a x)} \, dx,x,x^{1-n}\right )}{1-n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {n}{x}+\frac {a-a n}{b+a x}\right ) \, dx,x,x^{1-n}\right )}{1-n}\\ &=n \log (x)+\log \left (b+a x^{1-n}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 17, normalized size = 1.00 \[ \log \left (a x^{1-n}+b\right )+n \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 10, normalized size = 0.59 \[ \log \left (a x + b x^{n}\right ) \]
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 \[ \int \frac {b n x^{n - 1} + a}{{\left (a x^{-n + 1} + b\right )} x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 13, normalized size = 0.76 \[ \ln \left (a x +b \,{\mathrm e}^{n \ln \relax (x )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.95, size = 86, normalized size = 5.06 \[ b n {\left (\frac {\log \relax (x)}{b} - \frac {n \log \relax (x)}{b {\left (n - 1\right )}} + \frac {\log \left (\frac {a x + b x^{n}}{b}\right )}{b {\left (n - 1\right )}}\right )} + a {\left (\frac {n \log \relax (x)}{a {\left (n - 1\right )}} - \frac {\log \left (\frac {a x + b x^{n}}{b}\right )}{a {\left (n - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.37, size = 39, normalized size = 2.29 \[ -\frac {\ln \left (b+a\,x^{1-n}\right )-2\,n\,\mathrm {atanh}\left (\frac {2\,a\,x^{1-n}}{b}+1\right )}{n-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 53.07, size = 8, normalized size = 0.47 \[ \log {\left (a x + b x^{n} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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