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 = 0.18, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, 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 {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^3} \, 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 = 0.04, 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 = 0.46, size = 45, normalized size = 2.05 \[ -\frac {1}{2 \, {\left (2 \, {\left (n \log \relax (x) + \log \relax (c)\right )}^{q} a b x^{m} + {\left (n \log \relax (x) + \log \relax (c)\right )}^{2 \, q} b^{2} + a^{2} x^{2 \, m}\right )}} \]
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 \[ \int \frac {b n q \log \left (c x^{n}\right )^{q - 1} + a m x^{m}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.44, size = 68, normalized size = 3.09 \[ -\frac {1}{2 \left (a \,x^{m}+b \left (-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )^{q}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.67, size = 49, normalized size = 2.23 \[ -\frac {1}{2 \, {\left (a^{2} x^{2 \, m} + b^{2} {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{2 \, q} + 2 \, a b e^{\left (m \log \relax (x) + q \log \left (\log \relax (c) + \log \left (x^{n}\right )\right )\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {a\,m\,x^m+b\,n\,q\,{\ln \left (c\,x^n\right )}^{q-1}}{x\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^3} \,d x \]
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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