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 = 0.13, 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 = {2624}
\begin {gather*} -\frac {1}{2 \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2624
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 = 0.03, size = 22, normalized size = 1.00 \begin {gather*} -\frac {1}{2 \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 2.87, size = 68, normalized size = 3.09
method | result | size |
risch | \(-\frac {1}{2 \left (a \,x^{m}+b \left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i x^{n}\right )\right )}{2}\right )^{q}\right )^{2}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (20) = 40\).
time = 0.57, size = 49, normalized size = 2.23 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (20) = 40\).
time = 0.36, size = 45, normalized size = 2.05 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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