Optimal. Leaf size=82 \[ \frac {6 f^{a+\frac {b}{x}}}{b^4 \log ^4(f)}-\frac {6 f^{a+\frac {b}{x}}}{b^3 x \log ^3(f)}+\frac {3 f^{a+\frac {b}{x}}}{b^2 x^2 \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^3 \log (f)} \]
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Rubi [A] time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 2209
Rule 2212
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*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 0.65 \[ \frac {f^{a+\frac {b}{x}} \left (-b^3 \log ^3(f)+3 b^2 x \log ^2(f)-6 b x^2 \log (f)+6 x^3\right )}{b^4 x^3 \log ^4(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 55, normalized size = 0.67 \[ -\frac {{\left (b^{3} \log \relax (f)^{3} - 3 \, b^{2} x \log \relax (f)^{2} + 6 \, b x^{2} \log \relax (f) - 6 \, x^{3}\right )} f^{\frac {a x + b}{x}}}{b^{4} x^{3} \log \relax (f)^{4}} \]
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 \[ \int \frac {f^{a + \frac {b}{x}}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 96, normalized size = 1.17 \[ \frac {-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b \ln \relax (f )}+\frac {3 x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}-\frac {6 x^{3} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}+\frac {6 x^{4} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{4} \ln \relax (f )^{4}}}{x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.30, size = 21, normalized size = 0.26 \[ \frac {f^{a} \Gamma \left (4, -\frac {b \log \relax (f)}{x}\right )}{b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 57, normalized size = 0.70 \[ -\frac {f^{a+\frac {b}{x}}\,\left (\frac {1}{b\,\ln \relax (f)}+\frac {6\,x^2}{b^3\,{\ln \relax (f)}^3}-\frac {6\,x^3}{b^4\,{\ln \relax (f)}^4}-\frac {3\,x}{b^2\,{\ln \relax (f)}^2}\right )}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 53, normalized size = 0.65 \[ \frac {f^{a + \frac {b}{x}} \left (- b^{3} \log {\relax (f )}^{3} + 3 b^{2} x \log {\relax (f )}^{2} - 6 b x^{2} \log {\relax (f )} + 6 x^{3}\right )}{b^{4} x^{3} \log {\relax (f )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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