Optimal. Leaf size=24 \[ \frac{(a+b x) \log \left (c (a+b x)^p\right )}{b}-p x \]
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Rubi [A] time = 0.0090786, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2389, 2295} \[ \frac{(a+b x) \log \left (c (a+b x)^p\right )}{b}-p x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2295
Rubi steps
\begin{align*} \int \log \left (c (a+b x)^p\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x\right )}{b}\\ &=-p x+\frac{(a+b x) \log \left (c (a+b x)^p\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0056086, size = 24, normalized size = 1. \[ \frac{(a+b x) \log \left (c (a+b x)^p\right )}{b}-p x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 30, normalized size = 1.3 \begin{align*} \ln \left ( c \left ( bx+a \right ) ^{p} \right ) x-px+{\frac{ap\ln \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0432, size = 47, normalized size = 1.96 \begin{align*} -b p{\left (\frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}}\right )} + x \log \left ({\left (b x + a\right )}^{p} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90589, size = 73, normalized size = 3.04 \begin{align*} -\frac{b p x - b x \log \left (c\right ) -{\left (b p x + a p\right )} \log \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.539807, size = 37, normalized size = 1.54 \begin{align*} \begin{cases} \frac{a p \log{\left (a + b x \right )}}{b} + p x \log{\left (a + b x \right )} - p x + x \log{\left (c \right )} & \text{for}\: b \neq 0 \\x \log{\left (a^{p} c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18071, size = 53, normalized size = 2.21 \begin{align*} \frac{{\left (b x + a\right )} p \log \left (b x + a\right )}{b} - \frac{{\left (b x + a\right )} p}{b} + \frac{{\left (b x + a\right )} \log \left (c\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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