Optimal. Leaf size=31 \[ \frac {(a+b \log (c (e+f x)))^{1+p}}{b d f (1+p)} \]
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Rubi [A]
time = 0.06, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2437, 12, 2339,
30} \begin {gather*} \frac {(a+b \log (c (e+f x)))^{p+1}}{b d f (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2339
Rule 2437
Rubi steps
\begin {align*} \int \frac {(a+b \log (c (e+f x)))^p}{d e+d f x} \, dx &=\frac {\text {Subst}\left (\int \frac {(a+b \log (c x))^p}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {(a+b \log (c x))^p}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac {\text {Subst}\left (\int x^p \, dx,x,a+b \log (c (e+f x))\right )}{b d f}\\ &=\frac {(a+b \log (c (e+f x)))^{1+p}}{b d f (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.00 \begin {gather*} \frac {(a+b \log (c (e+f x)))^{1+p}}{b d f (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 32, normalized size = 1.03
method | result | size |
default | \(\frac {\left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{1+p}}{b d f \left (1+p \right )}\) | \(32\) |
norman | \(\frac {\ln \left (c \left (f x +e \right )\right ) {\mathrm e}^{p \ln \left (a +b \ln \left (c \left (f x +e \right )\right )\right )}}{d f \left (1+p \right )}+\frac {a \,{\mathrm e}^{p \ln \left (a +b \ln \left (c \left (f x +e \right )\right )\right )}}{b d f \left (1+p \right )}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 33, normalized size = 1.06 \begin {gather*} \frac {{\left (b \log \left (c f x + c e\right ) + a\right )}^{p + 1}}{b d f {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 43, normalized size = 1.39 \begin {gather*} \frac {{\left (b \log \left (c f x + c e\right ) + a\right )} {\left (b \log \left (c f x + c e\right ) + a\right )}^{p}}{b d f p + b d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e + f x}\, dx}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.00, size = 33, normalized size = 1.06 \begin {gather*} \frac {{\left (b \log \left (c f x + c e\right ) + a\right )}^{p + 1}}{b d f {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 31, normalized size = 1.00 \begin {gather*} \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^{p+1}}{b\,d\,f\,\left (p+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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