Optimal. Leaf size=65 \[ -\frac{f^{a+\frac{b}{x}} \left (12 b^2 x^2 \log ^2(f)-4 b^3 x \log ^3(f)+b^4 \log ^4(f)-24 b x^3 \log (f)+24 x^4\right )}{b^5 x^4 \log ^5(f)} \]
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Rubi [C] time = 0.0183533, antiderivative size = 22, normalized size of antiderivative = 0.34, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2218} \[ -\frac{f^a \text{Gamma}\left (5,-\frac{b \log (f)}{x}\right )}{b^5 \log ^5(f)} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int \frac{f^{a+\frac{b}{x}}}{x^6} \, dx &=-\frac{f^a \Gamma \left (5,-\frac{b \log (f)}{x}\right )}{b^5 \log ^5(f)}\\ \end{align*}
Mathematica [C] time = 0.0027207, size = 22, normalized size = 0.34 \[ -\frac{f^a \text{Gamma}\left (5,-\frac{b \log (f)}{x}\right )}{b^5 \log ^5(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 119, normalized size = 1.8 \begin{align*}{\frac{1}{{x}^{5}} \left ( -24\,{\frac{{x}^{5}}{{b}^{5} \left ( \ln \left ( f \right ) \right ) ^{5}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}+24\,{\frac{{x}^{4}}{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-12\,{\frac{{x}^{3}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}+4\,{\frac{{x}^{2}}{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-{\frac{x}{b\ln \left ( f \right ) }{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.28846, size = 30, normalized size = 0.46 \begin{align*} -\frac{f^{a} \Gamma \left (5, -\frac{b \log \left (f\right )}{x}\right )}{b^{5} \log \left (f\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8285, size = 162, normalized size = 2.49 \begin{align*} -\frac{{\left (b^{4} \log \left (f\right )^{4} - 4 \, b^{3} x \log \left (f\right )^{3} + 12 \, b^{2} x^{2} \log \left (f\right )^{2} - 24 \, b x^{3} \log \left (f\right ) + 24 \, x^{4}\right )} f^{\frac{a x + b}{x}}}{b^{5} x^{4} \log \left (f\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.155604, size = 66, normalized size = 1.02 \begin{align*} \frac{f^{a + \frac{b}{x}} \left (- b^{4} \log{\left (f \right )}^{4} + 4 b^{3} x \log{\left (f \right )}^{3} - 12 b^{2} x^{2} \log{\left (f \right )}^{2} + 24 b x^{3} \log{\left (f \right )} - 24 x^{4}\right )}{b^{5} x^{4} \log{\left (f \right )}^{5}} \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^{a + \frac{b}{x}}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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