Optimal. Leaf size=27 \[ x \log \left (c \left (a+\frac {b}{x}\right )^p\right )+\frac {b p \log (b+a x)}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2498, 269, 31}
\begin {gather*} x \log \left (c \left (a+\frac {b}{x}\right )^p\right )+\frac {b p \log (a x+b)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 269
Rule 2498
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*}
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Mathematica [A]
time = 0.00, size = 37, normalized size = 1.37 \begin {gather*} \frac {b p \log \left (a+\frac {b}{x}\right )}{a}+x \log \left (c \left (a+\frac {b}{x}\right )^p\right )+\frac {b p \log (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 30, normalized size = 1.11
method | result | size |
default | \(x \ln \left (c \left (\frac {a x +b}{x}\right )^{p}\right )+\frac {b p \ln \left (a x +b \right )}{a}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 27, normalized size = 1.00 \begin {gather*} x \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) + \frac {b p \log \left (a x + b\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 33, normalized size = 1.22 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 36, normalized size = 1.33 \begin {gather*} \begin {cases} x \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )} + \frac {b p \log {\left (a x + b \right )}}{a} & \text {for}\: a \neq 0 \\p x + x \log {\left (c \left (\frac {b}{x}\right )^{p} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 96 vs.
\(2 (27) = 54\).
time = 4.38, size = 96, normalized size = 3.56 \begin {gather*} -\frac {\frac {b^{2} p \log \left (-a + \frac {a x + b}{x}\right )}{a} + \frac {b^{2} p \log \left (\frac {a x + b}{x}\right )}{a - \frac {a x + b}{x}} - \frac {b^{2} p \log \left (\frac {a x + b}{x}\right )}{a} + \frac {b^{2} \log \left (c\right )}{a - \frac {a x + b}{x}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 27, normalized size = 1.00 \begin {gather*} x\,\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )+\frac {b\,p\,\ln \left (b+a\,x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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