Optimal. Leaf size=115 \[ \frac{i e^{-\frac{a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c (e+f x))}{b}\right )}{c d f^2}+\frac{(f h-e i) (a+b \log (c (e+f x)))^{p+1}}{b d f^2 (p+1)} \]
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Rubi [A] time = 0.28188, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {2411, 12, 2353, 2299, 2181, 2302, 30} \[ \frac{i e^{-\frac{a}{b}} (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c (e+f x))}{b}\right )}{c d f^2}+\frac{(f h-e i) (a+b \log (c (e+f x)))^{p+1}}{b d f^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 2411
Rule 12
Rule 2353
Rule 2299
Rule 2181
Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{(h+212 x) (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-212 e+f h}{f}+\frac{212 x}{f}\right ) (a+b \log (c x))^p}{d x} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{-212 e+f h}{f}+\frac{212 x}{f}\right ) (a+b \log (c x))^p}{x} \, dx,x,e+f x\right )}{d f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{212 (a+b \log (c x))^p}{f}+\frac{(-212 e+f h) (a+b \log (c x))^p}{f x}\right ) \, dx,x,e+f x\right )}{d f}\\ &=\frac{212 \operatorname{Subst}\left (\int (a+b \log (c x))^p \, dx,x,e+f x\right )}{d f^2}-\frac{(212 e-f h) \operatorname{Subst}\left (\int \frac{(a+b \log (c x))^p}{x} \, dx,x,e+f x\right )}{d f^2}\\ &=\frac{212 \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log (c (e+f x))\right )}{c d f^2}-\frac{(212 e-f h) \operatorname{Subst}\left (\int x^p \, dx,x,a+b \log (c (e+f x))\right )}{b d f^2}\\ &=-\frac{(212 e-f h) (a+b \log (c (e+f x)))^{1+p}}{b d f^2 (1+p)}+\frac{212 e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log (c (e+f x))}{b}\right ) (a+b \log (c (e+f x)))^p \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p}}{c d f^2}\\ \end{align*}
Mathematica [A] time = 0.2233, size = 106, normalized size = 0.92 \[ \frac{(a+b \log (c (e+f x)))^p \left (\frac{i e^{-\frac{a}{b}} \left (-\frac{a+b \log (c (e+f x))}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log (c (e+f x))}{b}\right )}{c}+\frac{(f h-e i) (a+b \log (c (e+f x)))}{b (p+1)}\right )}{d f^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.487, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ix+h \right ) \left ( a+b\ln \left ( c \left ( fx+e \right ) \right ) \right ) ^{p}}{dfx+de}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} i \int \frac{{\left (b \log \left (f x + e\right ) + b \log \left (c\right ) + a\right )}^{p} x}{d f x + d e}\,{d x} + \frac{{\left (b c \log \left (c f x + c e\right ) + a c\right )}{\left (b \log \left (c f x + c e\right ) + a\right )}^{p} h}{b c d f{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82375, size = 270, normalized size = 2.35 \begin{align*} \frac{{\left (b i p + b i\right )} e^{\left (-\frac{b p \log \left (-\frac{1}{b}\right ) + a}{b}\right )} \Gamma \left (p + 1, -\frac{b \log \left (c f x + c e\right ) + a}{b}\right ) +{\left (a c f h - a c e i +{\left (b c f h - b c e i\right )} \log \left (c f x + c e\right )\right )}{\left (b \log \left (c f x + c e\right ) + a\right )}^{p}}{b c d f^{2} p + b c d f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i x + h\right )}{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{d f x + d e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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