Optimal. Leaf size=68 \[ -\frac{b c \log (f) f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a (a+b x)}\right )}{a^2}-\frac{b f^{\frac{c}{a+b x}}}{a}-\frac{f^{\frac{c}{a+b x}}}{x} \]
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Rubi [A] time = 0.39663, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {2223, 6742, 2222, 2210, 2228, 2178, 2209} \[ -\frac{b c \log (f) f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a (a+b x)}\right )}{a^2}-\frac{b f^{\frac{c}{a+b x}}}{a}-\frac{f^{\frac{c}{a+b x}}}{x} \]
Antiderivative was successfully verified.
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Rule 2223
Rule 6742
Rule 2222
Rule 2210
Rule 2228
Rule 2178
Rule 2209
Rubi steps
\begin{align*} \int \frac{f^{\frac{c}{a+b x}}}{x^2} \, dx &=-\frac{f^{\frac{c}{a+b x}}}{x}-(b c \log (f)) \int \frac{f^{\frac{c}{a+b x}}}{x (a+b x)^2} \, dx\\ &=-\frac{f^{\frac{c}{a+b x}}}{x}-(b c \log (f)) \int \left (\frac{f^{\frac{c}{a+b x}}}{a^2 x}-\frac{b f^{\frac{c}{a+b x}}}{a (a+b x)^2}-\frac{b f^{\frac{c}{a+b x}}}{a^2 (a+b x)}\right ) \, dx\\ &=-\frac{f^{\frac{c}{a+b x}}}{x}-\frac{(b c \log (f)) \int \frac{f^{\frac{c}{a+b x}}}{x} \, dx}{a^2}+\frac{\left (b^2 c \log (f)\right ) \int \frac{f^{\frac{c}{a+b x}}}{a+b x} \, dx}{a^2}+\frac{\left (b^2 c \log (f)\right ) \int \frac{f^{\frac{c}{a+b x}}}{(a+b x)^2} \, dx}{a}\\ &=-\frac{b f^{\frac{c}{a+b x}}}{a}-\frac{f^{\frac{c}{a+b x}}}{x}-\frac{b c \text{Ei}\left (\frac{c \log (f)}{a+b x}\right ) \log (f)}{a^2}-\frac{(b c \log (f)) \int \frac{f^{\frac{c}{a+b x}}}{x (a+b x)} \, dx}{a}-\frac{\left (b^2 c \log (f)\right ) \int \frac{f^{\frac{c}{a+b x}}}{a+b x} \, dx}{a^2}\\ &=-\frac{b f^{\frac{c}{a+b x}}}{a}-\frac{f^{\frac{c}{a+b x}}}{x}-\frac{(b c \log (f)) \operatorname{Subst}\left (\int \frac{f^{\frac{c}{a}-\frac{b c x}{a}}}{x} \, dx,x,\frac{x}{a+b x}\right )}{a^2}\\ &=-\frac{b f^{\frac{c}{a+b x}}}{a}-\frac{f^{\frac{c}{a+b x}}}{x}-\frac{b c f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a (a+b x)}\right ) \log (f)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.102447, size = 68, normalized size = 1. \[ -\frac{b c \log (f) f^{\frac{c}{a}} \text{Ei}\left (-\frac{b c x \log (f)}{a^2+b x a}\right )}{a^2}-\frac{b f^{\frac{c}{a+b x}}}{a}-\frac{f^{\frac{c}{a+b x}}}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 80, normalized size = 1.2 \begin{align*}{\frac{cb\ln \left ( f \right ) }{{a}^{2}}{f}^{{\frac{c}{bx+a}}} \left ({\frac{c\ln \left ( f \right ) }{bx+a}}-{\frac{c\ln \left ( f \right ) }{a}} \right ) ^{-1}}+{\frac{cb\ln \left ( f \right ) }{{a}^{2}}{f}^{{\frac{c}{a}}}{\it Ei} \left ( 1,-{\frac{c\ln \left ( f \right ) }{bx+a}}+{\frac{c\ln \left ( f \right ) }{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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54048, size = 131, normalized size = 1.93 \begin{align*} -\frac{b c f^{\frac{c}{a}} x{\rm Ei}\left (-\frac{b c x \log \left (f\right )}{a b x + a^{2}}\right ) \log \left (f\right ) +{\left (a b x + a^{2}\right )} f^{\frac{c}{b x + a}}}{a^{2} x} \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^{2}}\, 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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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