Integrand size = 16, antiderivative size = 46 \[ \int (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {b n (f x)^{1+m}}{f (1+m)^2}+\frac {(f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)} \]
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Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2341} \[ \int (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {(f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}-\frac {b n (f x)^{m+1}}{f (m+1)^2} \]
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Rule 2341
Rubi steps \begin{align*} \text {integral}& = -\frac {b n (f x)^{1+m}}{f (1+m)^2}+\frac {(f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70 \[ \int (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x (f x)^m \left (a+a m-b n+b (1+m) \log \left (c x^n\right )\right )}{(1+m)^2} \]
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Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.48
method | result | size |
parallelrisch | \(-\frac {-x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b m -x \left (f x \right )^{m} \ln \left (c \,x^{n}\right ) b -x \left (f x \right )^{m} a m +x \left (f x \right )^{m} b n -x \left (f x \right )^{m} a}{\left (1+m \right )^{2}}\) | \(68\) |
risch | \(\frac {b x \,x^{m} f^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i f x \right ) \pi m \left (\operatorname {csgn}\left (i f x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i f x \right )+\operatorname {csgn}\left (i f \right )\right )}{2}} \ln \left (x^{n}\right )}{1+m}-\frac {\left (i \pi b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) m -i \pi b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} m -i \pi b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2} m +i \pi b \operatorname {csgn}\left (i c \,x^{n}\right )^{3} m +i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-2 b \ln \left (c \right ) m -2 b \ln \left (c \right )-2 a m +2 b n -2 a \right ) x \,x^{m} f^{m} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i f x \right ) \pi m \left (\operatorname {csgn}\left (i f x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i f x \right )+\operatorname {csgn}\left (i f \right )\right )}{2}}}{2 \left (1+m \right )^{2}}\) | \(305\) |
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Time = 0.31 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left ({\left (b m + b\right )} n x \log \left (x\right ) + {\left (b m + b\right )} x \log \left (c\right ) + {\left (a m - b n + a\right )} x\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{2} + 2 \, m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (37) = 74\).
Time = 2.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 3.07 \[ \int (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \frac {a m x \left (f x\right )^{m}}{m^{2} + 2 m + 1} + \frac {a x \left (f x\right )^{m}}{m^{2} + 2 m + 1} + \frac {b m x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{2} + 2 m + 1} - \frac {b n x \left (f x\right )^{m}}{m^{2} + 2 m + 1} + \frac {b x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{2} + 2 m + 1} & \text {for}\: m \neq -1 \\\frac {\begin {cases} a \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (- a - b \log {\left (c \right )}\right ) \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}}{f} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.24 \[ \int (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {b f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a}{f {\left (m + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (46) = 92\).
Time = 0.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.07 \[ \int (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {b f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac {b f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (f x\right )^{m} b x \log \left (c\right )}{m + 1} + \frac {\left (f x\right )^{m} a x}{m + 1} \]
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Timed out. \[ \int (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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