Optimal. Leaf size=34 \[ a x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-b p q x \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2389, 2295, 2445} \[ a x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-b p q x \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2445
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx &=a x+b \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx\\ &=a x+b \operatorname {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 \operatorname {Subst}\left (\frac {\operatorname {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*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \[ a x+\frac {b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-b p q x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 50, normalized size = 1.47 \[ \frac {b f q x \log \relax (d) + b f x \log \relax (c) - {\left (b f p q - a f\right )} x + {\left (b f p q x + b e p q\right )} \log \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 64, normalized size = 1.88 \[ {\left (\frac {{\left (f x + e\right )} p q \log \left (f x + e\right )}{f} - \frac {{\left (f x + e\right )} p q}{f} + \frac {{\left (f x + e\right )} q \log \relax (d)}{f} + \frac {{\left (f x + e\right )} \log \relax (c)}{f}\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 1.24 \[ \frac {b e p q \ln \left (f x +e \right )}{f}-b p q x +b x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 45, normalized size = 1.32 \[ -b f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + b x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 41, normalized size = 1.21 \[ x\,\left (a-b\,p\,q\right )+b\,x\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )+\frac {b\,e\,p\,q\,\ln \left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 58, normalized size = 1.71 \[ a x + b \left (\begin {cases} \frac {e p q \log {\left (e + f x \right )}}{f} + p q x \log {\left (e + f x \right )} - p q x + q x \log {\relax (d )} + x \log {\relax (c )} & \text {for}\: f \neq 0 \\x \log {\left (c \left (d e^{p}\right )^{q} \right )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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