Optimal. Leaf size=47 \[ -\frac{b^2 p \log (a x+b)}{2 a^2}+\frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x}{2 a} \]
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Rubi [A] time = 0.0210677, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2455, 193, 43} \[ -\frac{b^2 p \log (a x+b)}{2 a^2}+\frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x}{2 a} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 193
Rule 43
Rubi steps
\begin{align*} \int x \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx &=\frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{2} (b p) \int \frac{1}{a+\frac{b}{x}} \, dx\\ &=\frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{2} (b p) \int \frac{x}{b+a x} \, dx\\ &=\frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{2} (b p) \int \left (\frac{1}{a}-\frac{b}{a (b+a x)}\right ) \, dx\\ &=\frac{b p x}{2 a}+\frac{1}{2} x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )-\frac{b^2 p \log (b+a x)}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0168691, size = 40, normalized size = 0.85 \[ \frac{1}{2} \left (\frac{b p (a x-b \log (a x+b))}{a^2}+x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.081, size = 0, normalized size = 0. \begin{align*} \int x\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1689, size = 54, normalized size = 1.15 \begin{align*} \frac{1}{2} \, b p{\left (\frac{x}{a} - \frac{b \log \left (a x + b\right )}{a^{2}}\right )} + \frac{1}{2} \, x^{2} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12569, size = 116, normalized size = 2.47 \begin{align*} \frac{a^{2} p x^{2} \log \left (\frac{a x + b}{x}\right ) + a^{2} x^{2} \log \left (c\right ) + a b p x - b^{2} p \log \left (a x + b\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.21775, size = 82, normalized size = 1.74 \begin{align*} \begin{cases} \frac{p x^{2} \log{\left (a + \frac{b}{x} \right )}}{2} + \frac{x^{2} \log{\left (c \right )}}{2} + \frac{b p x}{2 a} - \frac{b^{2} p \log{\left (a x + b \right )}}{2 a^{2}} & \text{for}\: a \neq 0 \\\frac{p x^{2} \log{\left (b \right )}}{2} - \frac{p x^{2} \log{\left (x \right )}}{2} + \frac{p x^{2}}{4} + \frac{x^{2} \log{\left (c \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31848, size = 69, normalized size = 1.47 \begin{align*} \frac{1}{2} \, p x^{2} \log \left (a x + b\right ) - \frac{1}{2} \, p x^{2} \log \left (x\right ) + \frac{1}{2} \, x^{2} \log \left (c\right ) + \frac{b p x}{2 \, a} - \frac{b^{2} p \log \left (a x + b\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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