Optimal. Leaf size=27 \[ x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p \log (a x+b)}{a} \]
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Rubi [A] time = 0.0087283, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2448, 263, 31} \[ x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p \log (a x+b)}{a} \]
Antiderivative was successfully verified.
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Rule 2448
Rule 263
Rule 31
Rubi steps
\begin{align*} \int \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx &=x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+(b p) \int \frac{1}{\left (a+\frac{b}{x}\right ) x} \, dx\\ &=x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+(b p) \int \frac{1}{b+a x} \, dx\\ &=x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p \log (b+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.002082, size = 37, normalized size = 1.37 \[ x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p \log \left (a+\frac{b}{x}\right )}{a}+\frac{b p \log (x)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 30, normalized size = 1.1 \begin{align*} x\ln \left ( c \left ({\frac{ax+b}{x}} \right ) ^{p} \right ) +{\frac{bp\ln \left ( ax+b \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14417, size = 36, normalized size = 1.33 \begin{align*} x \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) + \frac{b p \log \left (a x + b\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20548, size = 81, normalized size = 3. \begin{align*} \frac{a p x \log \left (\frac{a x + b}{x}\right ) + b p \log \left (a x + b\right ) + a x \log \left (c\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.31805, size = 48, normalized size = 1.78 \begin{align*} \begin{cases} p x \log{\left (a + \frac{b}{x} \right )} + x \log{\left (c \right )} + \frac{b p \log{\left (a x + b \right )}}{a} & \text{for}\: a \neq 0 \\p x \log{\left (b \right )} - p x \log{\left (x \right )} + p x + x \log{\left (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 = 1.27674, size = 43, normalized size = 1.59 \begin{align*} p x \log \left (a x + b\right ) - p x \log \left (x\right ) + \frac{b p \log \left (a x + b\right )}{a} + x \log \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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