Optimal. Leaf size=82 \[ \frac{3 f^{a+\frac{b}{x}}}{b^2 x^2 \log ^2(f)}-\frac{6 f^{a+\frac{b}{x}}}{b^3 x \log ^3(f)}+\frac{6 f^{a+\frac{b}{x}}}{b^4 \log ^4(f)}-\frac{f^{a+\frac{b}{x}}}{b x^3 \log (f)} \]
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Rubi [A] time = 0.0815529, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ \frac{3 f^{a+\frac{b}{x}}}{b^2 x^2 \log ^2(f)}-\frac{6 f^{a+\frac{b}{x}}}{b^3 x \log ^3(f)}+\frac{6 f^{a+\frac{b}{x}}}{b^4 \log ^4(f)}-\frac{f^{a+\frac{b}{x}}}{b x^3 \log (f)} \]
Antiderivative was successfully verified.
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Rule 2212
Rule 2209
Rubi steps
\begin{align*} \int \frac{f^{a+\frac{b}{x}}}{x^5} \, dx &=-\frac{f^{a+\frac{b}{x}}}{b x^3 \log (f)}-\frac{3 \int \frac{f^{a+\frac{b}{x}}}{x^4} \, dx}{b \log (f)}\\ &=\frac{3 f^{a+\frac{b}{x}}}{b^2 x^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x^3 \log (f)}+\frac{6 \int \frac{f^{a+\frac{b}{x}}}{x^3} \, dx}{b^2 \log ^2(f)}\\ &=-\frac{6 f^{a+\frac{b}{x}}}{b^3 x \log ^3(f)}+\frac{3 f^{a+\frac{b}{x}}}{b^2 x^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x^3 \log (f)}-\frac{6 \int \frac{f^{a+\frac{b}{x}}}{x^2} \, dx}{b^3 \log ^3(f)}\\ &=\frac{6 f^{a+\frac{b}{x}}}{b^4 \log ^4(f)}-\frac{6 f^{a+\frac{b}{x}}}{b^3 x \log ^3(f)}+\frac{3 f^{a+\frac{b}{x}}}{b^2 x^2 \log ^2(f)}-\frac{f^{a+\frac{b}{x}}}{b x^3 \log (f)}\\ \end{align*}
Mathematica [A] time = 0.010215, size = 53, normalized size = 0.65 \[ \frac{f^{a+\frac{b}{x}} \left (3 b^2 x \log ^2(f)-b^3 \log ^3(f)-6 b x^2 \log (f)+6 x^3\right )}{b^4 x^3 \log ^4(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 96, normalized size = 1.2 \begin{align*}{\frac{1}{{x}^{4}} \left ( 6\,{\frac{{x}^{4}}{{b}^{4} \left ( \ln \left ( f \right ) \right ) ^{4}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}-6\,{\frac{{x}^{3}}{ \left ( \ln \left ( f \right ) \right ) ^{3}{b}^{3}}{{\rm e}^{ \left ( a+{\frac{b}{x}} \right ) \ln \left ( f \right ) }}}+3\,{\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 = 1.22899, size = 28, normalized size = 0.34 \begin{align*} \frac{f^{a} \Gamma \left (4, -\frac{b \log \left (f\right )}{x}\right )}{b^{4} \log \left (f\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69579, size = 130, normalized size = 1.59 \begin{align*} -\frac{{\left (b^{3} \log \left (f\right )^{3} - 3 \, b^{2} x \log \left (f\right )^{2} + 6 \, b x^{2} \log \left (f\right ) - 6 \, x^{3}\right )} f^{\frac{a x + b}{x}}}{b^{4} x^{3} \log \left (f\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.145166, size = 53, normalized size = 0.65 \begin{align*} \frac{f^{a + \frac{b}{x}} \left (- b^{3} \log{\left (f \right )}^{3} + 3 b^{2} x \log{\left (f \right )}^{2} - 6 b x^{2} \log{\left (f \right )} + 6 x^{3}\right )}{b^{4} x^{3} \log{\left (f \right )}^{4}} \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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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