Integrand size = 14, antiderivative size = 29 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=a x-b n x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e} \]
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Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2436, 2332} \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=a x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e}-b n x \]
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Rule 2332
Rule 2436
Rubi steps \begin{align*} \text {integral}& = a x+b \int \log \left (c (d+e x)^n\right ) \, dx \\ & = a x+\frac {b \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e} \\ & = a x-b n x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=a x-b n x+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e} \]
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Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24
method | result | size |
default | \(a x +b \ln \left (c \left (e x +d \right )^{n}\right ) x -b n x +\frac {b n d \ln \left (e x +d \right )}{e}\) | \(36\) |
parts | \(a x +b \ln \left (c \left (e x +d \right )^{n}\right ) x -b n x +\frac {b n d \ln \left (e x +d \right )}{e}\) | \(36\) |
norman | \(\left (-b n +a \right ) x +b x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )+\frac {b n d \ln \left (e x +d \right )}{e}\) | \(38\) |
parallelrisch | \(\frac {b \left (x \ln \left (c \left (e x +d \right )^{n}\right ) d e n -x d e \,n^{2}+\ln \left (c \left (e x +d \right )^{n}\right ) d^{2} n \right )}{d e n}+a x\) | \(55\) |
risch | \(a x +b x \ln \left (\left (e x +d \right )^{n}\right )-\frac {i b \pi x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )}{2}+\frac {i b \pi x \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}+\frac {i b \pi x \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i b \pi x \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {b n d \ln \left (e x +d \right )}{e}+\ln \left (c \right ) b x -b n x\) | \(149\) |
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Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {b e x \log \left (c\right ) - {\left (b e n - a e\right )} x + {\left (b e n x + b d n\right )} \log \left (e x + d\right )}{e} \]
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Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=a x + b \left (\begin {cases} \frac {d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - n x + x \log {\left (c \left (d + e x\right )^{n} \right )} & \text {for}\: e \neq 0 \\x \log {\left (c d^{n} \right )} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-b e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + b x \log \left ({\left (e x + d\right )}^{n} c\right ) + a x \]
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Time = 0.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx={\left (\frac {{\left (e x + d\right )} n \log \left (e x + d\right )}{e} - \frac {{\left (e x + d\right )} n}{e} + \frac {{\left (e x + d\right )} \log \left (c\right )}{e}\right )} b + a x \]
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Time = 1.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=x\,\left (a-b\,n\right )+b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )+\frac {b\,d\,n\,\ln \left (d+e\,x\right )}{e} \]
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