Optimal. Leaf size=24 \[ -p x+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2436, 2332}
\begin {gather*} \frac {(a+b x) \log \left (c (a+b x)^p\right )}{b}-p x \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2436
Rubi steps
\begin {align*} \int \log \left (c (a+b x)^p\right ) \, dx &=\frac {\text {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*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} -p x+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 36, normalized size = 1.50
method | result | size |
norman | \(x \ln \left (c \,{\mathrm e}^{p \ln \left (b x +a \right )}\right )+\frac {p a \ln \left (b x +a \right )}{b}-p x\) | \(32\) |
default | \(\ln \left (c \left (b x +a \right )^{p}\right ) x -p b \left (\frac {x}{b}-\frac {a \ln \left (b x +a \right )}{b^{2}}\right )\) | \(36\) |
risch | \(x \ln \left (\left (b x +a \right )^{p}\right )-\frac {i \pi x \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{2}+\frac {i \pi x \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}+x \ln \left (c \right )+\frac {p a \ln \left (b x +a \right )}{b}-p x\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 35, normalized size = 1.46 \begin {gather*} -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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 32, normalized size = 1.33 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 36, normalized size = 1.50 \begin {gather*} \begin {cases} \frac {a \log {\left (c \left (a + b x\right )^{p} \right )}}{b} - p x + x \log {\left (c \left (a + b x\right )^{p} \right )} & \text {for}\: b \neq 0 \\x \log {\left (a^{p} c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.56, size = 39, normalized size = 1.62 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 29, normalized size = 1.21 \begin {gather*} x\,\ln \left (c\,{\left (a+b\,x\right )}^p\right )-p\,x+\frac {a\,p\,\ln \left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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