Optimal. Leaf size=39 \[ \frac {f^{a+\frac {b}{x}}}{b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x \log (f)} \]
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Rubi [A] time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ \frac {f^{a+\frac {b}{x}}}{b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x \log (f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx &=-\frac {f^{a+\frac {b}{x}}}{b x \log (f)}-\frac {\int \frac {f^{a+\frac {b}{x}}}{x^2} \, dx}{b \log (f)}\\ &=\frac {f^{a+\frac {b}{x}}}{b^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.69 \[ \frac {f^{a+\frac {b}{x}} (x-b \log (f))}{b^2 x \log ^2(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 31, normalized size = 0.79 \[ -\frac {{\left (b \log \relax (f) - x\right )} f^{\frac {a x + b}{x}}}{b^{2} x \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 49, normalized size = 1.26 \[ \frac {-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b \ln \relax (f )}+\frac {x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}}{x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.29, size = 21, normalized size = 0.54 \[ \frac {f^{a} \Gamma \left (2, -\frac {b \log \relax (f)}{x}\right )}{b^{2} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 27, normalized size = 0.69 \[ \frac {f^{a+\frac {b}{x}}\,\left (x-b\,\ln \relax (f)\right )}{b^2\,x\,{\ln \relax (f)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 22, normalized size = 0.56 \[ \frac {f^{a + \frac {b}{x}} \left (- b \log {\relax (f )} + x\right )}{b^{2} x \log {\relax (f )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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