Optimal. Leaf size=29 \[ \frac {(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-p q x \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2389, 2295, 2445} \[ \frac {(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-p q x \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rule 2445
Rubi steps
\begin {align*} \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx &=\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 )\\ &=\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 )\\ &=-p q x+\frac {(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 = 29, normalized size = 1.00 \[ \frac {(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-p q x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 42, normalized size = 1.45 \[ -\frac {f p q x - f q x \log \relax (d) - f x \log \relax (c) - {\left (f p q x + e p q\right )} \log \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 58, normalized size = 2.00 \[ \frac {{\left (f x + e\right )} p q \log \left (f x + e\right )}{f} - \frac {{\left (f x + e\right )} p q}{f} + \frac {{\left (f x + e\right )} q \log \relax (d)}{f} + \frac {{\left (f x + e\right )} \log \relax (c)}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 1.24 \[ \frac {e p q \ln \left (f x +e \right )}{f}-p q x +x \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 40, normalized size = 1.38 \[ -f p q {\left (\frac {x}{f} - \frac {e \log \left (f x + e\right )}{f^{2}}\right )} + x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 36, normalized size = 1.24 \[ x\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )+\frac {p\,q\,\left (e\,\ln \left (e+f\,x\right )-f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.95, size = 53, normalized size = 1.83 \[ \begin {cases} \frac {e p q \log {\left (e + f x \right )}}{f} + p q x \log {\left (e + f x \right )} - p q x + q x \log {\relax (d )} + x \log {\relax (c )} & \text {for}\: f \neq 0 \\x \log {\left (c \left (d e^{p}\right )^{q} \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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