Optimal. Leaf size=31 \[ \frac {(a+b \log (c (e+f x)))^{p+1}}{b d f (p+1)} \]
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Rubi [A] time = 0.08, 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 = {2390, 12, 2302, 30} \[ \frac {(a+b \log (c (e+f x)))^{p+1}}{b d f (p+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2302
Rule 2390
Rubi steps
\begin {align*} \int \frac {(a+b \log (c (e+f x)))^p}{d e+d f x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b \log (c x))^p}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+b \log (c x))^p}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac {\operatorname {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 \[ \frac {(a+b \log (c (e+f x)))^{p+1}}{b d f (p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 41, normalized size = 1.32 \[ \frac {{\left (b \log \left (c f x + c e\right ) + a\right )} {\left (b \log \left (c f x + c e\right ) + a\right )}^{p}}{b d f p + b d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 31, normalized size = 1.00 \[ \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p + 1}}{{\left (b p + b\right )} d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 1.03 \[ \frac {\left (b \ln \left (\left (f x +e \right ) c \right )+a \right )^{p +1}}{\left (p +1\right ) b d f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 32, normalized size = 1.03 \[ \frac {{\left (b \log \left (c f x + c e\right ) + a\right )}^{p + 1}}{b d f {\left (p + 1\right )}} \]
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 \[ \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^{p+1}}{b\,d\,f\,\left (p+1\right )} \]
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 \[ \frac {\int \frac {\left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e + f x}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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