Optimal. Leaf size=71 \[ \frac{i e^{-\frac{a}{b}} \text{Ei}\left (\frac{a+b \log (c (e+f x))}{b}\right )}{b c d f^2}+\frac{(f h-e i) \log (a+b \log (c (e+f x)))}{b d f^2} \]
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Rubi [A] time = 0.21831, antiderivative size = 71, 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, 2178, 2302, 29} \[ \frac{i e^{-\frac{a}{b}} \text{Ei}\left (\frac{a+b \log (c (e+f x))}{b}\right )}{b c d f^2}+\frac{(f h-e i) \log (a+b \log (c (e+f x)))}{b d f^2} \]
Antiderivative was successfully verified.
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Rule 2411
Rule 12
Rule 2353
Rule 2299
Rule 2178
Rule 2302
Rule 29
Rubi steps
\begin{align*} \int \frac{h+194 x}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\frac{-194 e+f h}{f}+\frac{194 x}{f}}{d x (a+b \log (c x))} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\frac{-194 e+f h}{f}+\frac{194 x}{f}}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{194}{f (a+b \log (c x))}+\frac{-194 e+f h}{f x (a+b \log (c x))}\right ) \, dx,x,e+f x\right )}{d f}\\ &=\frac{194 \operatorname{Subst}\left (\int \frac{1}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^2}-\frac{(194 e-f h) \operatorname{Subst}\left (\int \frac{1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f^2}\\ &=\frac{194 \operatorname{Subst}\left (\int \frac{e^x}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c d f^2}-\frac{(194 e-f h) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d f^2}\\ &=\frac{194 e^{-\frac{a}{b}} \text{Ei}\left (\frac{a+b \log (c (e+f x))}{b}\right )}{b c d f^2}-\frac{(194 e-f h) \log (a+b \log (c (e+f x)))}{b d f^2}\\ \end{align*}
Mathematica [A] time = 0.14342, size = 76, normalized size = 1.07 \[ \frac{i e^{-\frac{a}{b}} \text{Ei}\left (\frac{a}{b}+\log (c (e+f x))\right )+c f h \log (f (a+b \log (c (e+f x))))-c e i \log (a+b \log (c (e+f x)))}{b c d f^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.529, size = 0, normalized size = 0. \begin{align*} \int{\frac{ix+h}{ \left ( dfx+de \right ) \left ( a+b\ln \left ( c \left ( fx+e \right ) \right ) \right ) }}\, 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{x}{b d e \log \left (c\right ) + a d e +{\left (b d f \log \left (c\right ) + a d f\right )} x +{\left (b d f x + b d e\right )} \log \left (f x + e\right )}\,{d x} + \frac{h \log \left (\frac{b \log \left (f x + e\right ) + b \log \left (c\right ) + a}{b}\right )}{b d f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68376, size = 157, normalized size = 2.21 \begin{align*} \frac{{\left ({\left (c f h - c e i\right )} e^{\frac{a}{b}} \log \left (b \log \left (c f x + c e\right ) + a\right ) + i \logintegral \left ({\left (c f x + c e\right )} e^{\frac{a}{b}}\right )\right )} e^{\left (-\frac{a}{b}\right )}}{b c d f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{h}{a e + a f x + b e \log{\left (c e + c f x \right )} + b f x \log{\left (c e + c f x \right )}}\, dx + \int \frac{i x}{a e + a f x + b e \log{\left (c e + c f x \right )} + b f x \log{\left (c e + c f x \right )}}\, dx}{d} \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{i x + h}{{\left (d f x + d e\right )}{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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