Optimal. Leaf size=10 \[ \log \left (a x+b x^n\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 17, normalized size of antiderivative = 1.70, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1593, 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
Rule 1593
Rubi steps
\begin {align*} \int \frac {a+b n x^{-1+n}}{a x+b x^n} \, dx &=\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 = 0.03, size = 17, normalized size = 1.70 \[ \log \left (a x^{1-n}+b\right )+n \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 10, normalized size = 1.00 \[ \log \left (a x + b x^{n}\right ) \]
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 \[ \int \frac {b n x^{n - 1} + a}{a x + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 13, normalized size = 1.30 \[ \ln \left (a x +b \,{\mathrm e}^{n \ln \relax (x )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 10, normalized size = 1.00 \[ \log \left (a x + b x^{n}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.27, size = 10, normalized size = 1.00 \[ \ln \left (b\,x^n+a\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.96, size = 32, normalized size = 3.20 \[ \begin {cases} \log {\left (x + \frac {b x^{n}}{a} \right )} & \text {for}\: a \neq 0 \\n \left (\frac {n^{2} \log {\relax (x )}}{n^{2} - n} - \frac {n \log {\relax (x )}}{n^{2} - n}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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