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 = 0.180028, antiderivative size = 22, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0401071, size = 22, normalized size = 1. \[ -\frac{1}{2 \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.184, size = 68, normalized size = 3.1 \begin{align*} -{\frac{1}{2\, \left ( a{x}^{m}+b \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -i/2\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) ^{q} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.132, size = 66, normalized size = 3. \begin{align*} -\frac{1}{2 \,{\left (a^{2} x^{2 \, m} + b^{2}{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{2 \, q} + 2 \, a b e^{\left (m \log \left (x\right ) + q \log \left (\log \left (c\right ) + \log \left (x^{n}\right )\right )\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04863, size = 116, normalized size = 5.27 \begin{align*} -\frac{1}{2 \,{\left (2 \,{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} a b x^{m} +{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{2 \, q} b^{2} + a^{2} x^{2 \, m}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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