Integrand size = 43, antiderivative size = 17 \[ \int \frac {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \]
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Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2621} \[ \int \frac {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \]
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Rule 2621
Rubi steps \begin{align*} \text {integral}& = \log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.58 (sec) , antiderivative size = 213, normalized size of antiderivative = 12.53
method | result | size |
risch | \(q \ln \left (\ln \left (x^{n}\right )-\frac {i \left (\pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )-\pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 i \ln \left (c \right )\right )}{2}\right )-q \ln \left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )+\ln \left ({\left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )}^{q}+\frac {a \,x^{m}}{b}\right )\) | \(213\) |
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none
Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\log \left ({\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}\right ) \]
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Timed out. \[ \int \frac {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\text {Timed out} \]
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none
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \frac {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\log \left (\frac {a x^{m} + b {\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}}{b}\right ) \]
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\[ \int \frac {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int { \frac {b n q \log \left (c x^{n}\right )^{q - 1} + a m x^{m}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x} \,d x } \]
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Timed out. \[ \int \frac {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx=\int \frac {a\,m\,x^m+b\,n\,q\,{\ln \left (c\,x^n\right )}^{q-1}}{x\,\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )} \,d x \]
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