Integrand size = 18, antiderivative size = 34 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=a x-b p q x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f} \]
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Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2436, 2332, 2495} \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=a x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-b p q x \]
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Rule 2332
Rule 2436
Rule 2495
Rubi steps \begin{align*} \text {integral}& = a x+b \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx \\ & = a x+b \text {Subst}\left (\int \log \left (c d^q (e+f x)^{p q}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = a x+b \text {Subst}\left (\frac {\text {Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = a x-b p q x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=a x-b p q x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f} \]
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Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24
method | result | size |
default | \(a x +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) x -b p q x +\frac {b q p e \ln \left (f x +e \right )}{f}\) | \(42\) |
parts | \(a x +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) x -b p q x +\frac {b q p e \ln \left (f x +e \right )}{f}\) | \(42\) |
parallelrisch | \(\frac {b \left (2 \ln \left (f x +e \right ) e^{2} p q -x e f p q +x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) e f -\ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) e^{2}\right )}{e f}+a x\) | \(71\) |
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Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=\frac {b f q x \log \left (d\right ) + b f x \log \left (c\right ) - {\left (b f p q - a f\right )} x + {\left (b f p q x + b e p q\right )} \log \left (f x + e\right )}{f} \]
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Time = 0.21 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=a x + b \left (\begin {cases} \frac {e \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{f} - p q x + x \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )} & \text {for}\: f \neq 0 \\x \log {\left (c \left (d e^{p}\right )^{q} \right )} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=-b f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + b x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a x \]
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Time = 0.32 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx={\left (\frac {{\left (f x + e\right )} p q \log \left (f x + e\right )}{f} - \frac {{\left (f x + e\right )} p q}{f} + \frac {{\left (f x + e\right )} q \log \left (d\right )}{f} + \frac {{\left (f x + e\right )} \log \left (c\right )}{f}\right )} b + a x \]
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Time = 1.35 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx=x\,\left (a-b\,p\,q\right )+b\,x\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )+\frac {b\,e\,p\,q\,\ln \left (e+f\,x\right )}{f} \]
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