Optimal. Leaf size=26 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^{p+1}}{b n (p+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2302, 30} \[ \frac {\left (a+b \log \left (c x^n\right )\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x} \, dx &=\frac {\operatorname {Subst}\left (\int x^p \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^{1+p}}{b n (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 35, normalized size = 1.35 \[ \frac {{\left (b n \log \relax (x) + b \log \relax (c) + a\right )} {\left (b n \log \relax (x) + b \log \relax (c) + a\right )}^{p}}{b n p + b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 27, normalized size = 1.04 \[ \frac {{\left (b n \log \relax (x) + b \log \relax (c) + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 27, normalized size = 1.04 \[ \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{p +1}}{\left (p +1\right ) b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 26, normalized size = 1.00 \[ \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.67, size = 26, normalized size = 1.00 \[ \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^{p+1}}{b\,n\,\left (p+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.86, size = 56, normalized size = 2.15 \[ - \begin {cases} - a^{p} \log {\relax (x )} & \text {for}\: b = 0 \\- \left (a + b \log {\relax (c )}\right )^{p} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {\begin {cases} \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b \log {\left (c x^{n} \right )} \right )} & \text {otherwise} \end {cases}}{b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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