Optimal. Leaf size=22 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
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Rubi [A] time = 0.02, antiderivative size = 22, 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} \[ \frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
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 )^2}{x} \, dx &=\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 51, normalized size = 2.32 \[ \frac {1}{3} \, b^{2} n^{2} \log \relax (x)^{3} + {\left (b^{2} n \log \relax (c) + a b n\right )} \log \relax (x)^{2} + {\left (b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c) + a^{2}\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 56, normalized size = 2.55 \[ \frac {1}{3} \, b^{2} n^{2} \log \relax (x)^{3} + b^{2} n \log \relax (c) \log \relax (x)^{2} + b^{2} \log \relax (c)^{2} \log \relax (x) + a b n \log \relax (x)^{2} + 2 \, a b \log \relax (c) \log \relax (x) + a^{2} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 56, normalized size = 2.55 \[ \frac {b^{2} \ln \left (c \,x^{n}\right )^{3}}{3 n}+\frac {a b \ln \left (c \,x^{n}\right )^{2}}{n}+\frac {a^{2} \ln \left (c \,x^{n}\right )}{n}+\frac {a^{3}}{3 b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 20, normalized size = 0.91 \[ \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{3 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.42, size = 37, normalized size = 1.68 \[ a^2\,\ln \relax (x)+\frac {b^2\,{\ln \left (c\,x^n\right )}^3}{3\,n}+\frac {a\,b\,{\ln \left (c\,x^n\right )}^2}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.81, size = 60, normalized size = 2.73 \[ \begin {cases} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\relax (c )} + b^{2} \log {\relax (c )}^{2}\right ) \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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