Optimal. Leaf size=22 \[ \frac {1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2 \]
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Rubi [A] time = 0.11, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2544} \[ \frac {1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2 \]
Antiderivative was successfully verified.
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Rule 2544
Rubi steps
\begin {align*} \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx &=\frac {1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 1.00 \[ \frac {1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 42, normalized size = 1.91 \[ {\left (n \log \relax (x) + \log \relax (c)\right )}^{q} a b x^{m} + \frac {1}{2} \, {\left (n \log \relax (x) + \log \relax (c)\right )}^{2 \, q} b^{2} + \frac {1}{2} \, a^{2} x^{2 \, m} \]
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 \[ \int \frac {{\left (b n q \log \left (c x^{n}\right )^{q - 1} + a m x^{m}\right )} {\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.67, size = 135, normalized size = 6.14 \[ a b \,x^{m} \left (-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )^{q}+\frac {a^{2} x^{2 m}}{2}+\frac {b^{2} \left (-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )^{2 q}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 20, normalized size = 0.91 \[ \frac {{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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