Optimal. Leaf size=20 \[ \frac {1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2 \]
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Rubi [A] time = 0.09, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2561, 2544} \[ \frac {1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2 \]
Antiderivative was successfully verified.
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Rule 2544
Rule 2561
Rubi steps
\begin {align*} \int \left (\frac {a}{x}+\frac {2 b n \log \left (c x^n\right )}{x^2}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right ) \, dx &=\int \frac {\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right )}{x^2} \, dx\\ &=\int \frac {\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x+b \log ^2\left (c x^n\right )\right )}{x} \, dx\\ &=\frac {1}{2} \left (a x+b \log ^2\left (c x^n\right )\right )^2\\ \end {align*}
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Mathematica [A] time = 0.00, size = 38, normalized size = 1.90 \[ \frac {a^2 x^2}{2}+a b x \log ^2\left (c x^n\right )+\frac {1}{2} b^2 \log ^4\left (c x^n\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 89, normalized size = 4.45 \[ \frac {1}{2} \, b^{2} n^{4} \log \relax (x)^{4} + 2 \, b^{2} n^{3} \log \relax (c) \log \relax (x)^{3} + a b x \log \relax (c)^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (3 \, b^{2} n^{2} \log \relax (c)^{2} + a b n^{2} x\right )} \log \relax (x)^{2} + 2 \, {\left (b^{2} n \log \relax (c)^{3} + a b n x \log \relax (c)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 90, normalized size = 4.50 \[ \frac {1}{2} \, b^{2} n^{4} \log \relax (x)^{4} + 2 \, b^{2} n^{3} \log \relax (c) \log \relax (x)^{3} + 2 \, b^{2} n \log \relax (c)^{3} \log \relax (x) + 2 \, a b n x \log \relax (c) \log \relax (x) + a b x \log \relax (c)^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (3 \, b^{2} n^{2} \log \relax (c)^{2} + a b n^{2} x\right )} \log \relax (x)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 63, normalized size = 3.15 \[ \frac {b^{2} \ln \left (c \,x^{n}\right )^{4}}{2}+2 a b n x \ln \left (c \,x^{n}\right )-2 a b n x \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )+a b x \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )^{2}+\frac {a^{2} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 74, normalized size = 3.70 \[ \frac {1}{2} \, b^{2} \log \left (c x^{n}\right )^{4} - 2 \, a b n^{2} x + 2 \, a b n x \log \left (c x^{n}\right ) + a b x \log \left (c x^{n}\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} a b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 18, normalized size = 0.90 \[ \frac {{\left (b\,{\ln \left (c\,x^n\right )}^2+a\,x\right )}^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.07, size = 117, normalized size = 5.85 \[ \frac {a^{2} x^{2}}{2} + a b n^{2} x \log {\relax (x )}^{2} - 2 a b n^{2} x \log {\relax (x )} + 2 a b n x \log {\relax (c )} \log {\relax (x )} - 2 a b n x \log {\relax (c )} + 2 a b n x \log {\left (c x^{n} \right )} + a b x \log {\relax (c )}^{2} - 2 b^{2} n \left (\begin {cases} - \log {\relax (c )}^{3} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{4}}{4 n} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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