Optimal. Leaf size=41 \[ f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a (a+b x)}\right )-\text{Ei}\left (\frac{c \log (f)}{a+b x}\right ) \]
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Rubi [A] time = 0.131148, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2222, 2210, 2228, 2178} \[ f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a (a+b x)}\right )-\text{Ei}\left (\frac{c \log (f)}{a+b x}\right ) \]
Antiderivative was successfully verified.
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Rule 2222
Rule 2210
Rule 2228
Rule 2178
Rubi steps
\begin{align*} \int \frac{f^{\frac{c}{a+b x}}}{x} \, dx &=a \int \frac{f^{\frac{c}{a+b x}}}{x (a+b x)} \, dx+b \int \frac{f^{\frac{c}{a+b x}}}{a+b x} \, dx\\ &=-\text{Ei}\left (\frac{c \log (f)}{a+b x}\right )+\operatorname{Subst}\left (\int \frac{f^{\frac{c}{a}-\frac{b c x}{a}}}{x} \, dx,x,\frac{x}{a+b x}\right )\\ &=-\text{Ei}\left (\frac{c \log (f)}{a+b x}\right )+f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a (a+b x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0309776, size = 41, normalized size = 1. \[ f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a^2+b x a}\right )-\text{Ei}\left (\frac{c \log (f)}{a+b x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.112, size = 47, normalized size = 1.2 \begin{align*} -{f}^{{\frac{c}{a}}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}}+{\frac{c\ln \left ( f \right ) }{a}} \right ) +{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}} \right ) \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*} \int \frac{f^{\frac{c}{b x + a}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56656, size = 89, normalized size = 2.17 \begin{align*} f^{\frac{c}{a}}{\rm Ei}\left (-\frac{b c x \log \left (f\right )}{a b x + a^{2}}\right ) -{\rm Ei}\left (\frac{c \log \left (f\right )}{b x + a}\right ) \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*} \int \frac{f^{\frac{c}{a + b x}}}{x}\, dx \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{f^{\frac{c}{b x + a}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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