Optimal. Leaf size=24 \[ -n x+\frac {(d+e x) \log \left (c (d+e x)^n\right )}{e} \]
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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 {(d+e x) \log \left (c (d+e x)^n\right )}{e}-n x \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2436
Rubi steps
\begin {align*} \int \log \left (c (d+e x)^n\right ) \, dx &=\frac {\text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}\\ &=-n x+\frac {(d+e x) \log \left (c (d+e x)^n\right )}{e}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} -n x+\frac {(d+e x) \log \left (c (d+e x)^n\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 36, normalized size = 1.50
method | result | size |
norman | \(x \ln \left (c \,{\mathrm e}^{n \ln \left (e x +d \right )}\right )+\frac {n d \ln \left (e x +d \right )}{e}-n x\) | \(32\) |
default | \(\ln \left (c \left (e x +d \right )^{n}\right ) x -e n \left (\frac {x}{e}-\frac {d \ln \left (e x +d \right )}{e^{2}}\right )\) | \(36\) |
risch | \(x \ln \left (\left (e x +d \right )^{n}\right )-\frac {i \pi x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}}{2}+\frac {n d \ln \left (e x +d \right )}{e}+x \ln \left (c \right )-n x\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 35, normalized size = 1.46 \begin {gather*} {\left (d e^{\left (-2\right )} \log \left (x e + d\right ) - x e^{\left (-1\right )}\right )} n e + x \log \left ({\left (x e + d\right )}^{n} c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 35, normalized size = 1.46 \begin {gather*} -{\left (n x e - x e \log \left (c\right ) - {\left (n x e + d n\right )} \log \left (x e + d\right )\right )} e^{\left (-1\right )} \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 {d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - n x + x \log {\left (c \left (d + e x\right )^{n} \right )} & \text {for}\: e \neq 0 \\x \log {\left (c d^{n} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.67, size = 40, normalized size = 1.67 \begin {gather*} {\left (x e + d\right )} n e^{\left (-1\right )} \log \left (x e + d\right ) - {\left (x e + d\right )} n e^{\left (-1\right )} + {\left (x e + d\right )} e^{\left (-1\right )} \log \left (c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 29, normalized size = 1.21 \begin {gather*} x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )-n\,x+\frac {d\,n\,\ln \left (d+e\,x\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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