Optimal. Leaf size=61 \[ -\frac {2 f^{a+\frac {b}{x}}}{b^3 \log ^3(f)}+\frac {2 f^{a+\frac {b}{x}}}{b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^2 \log (f)} \]
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Rubi [A] time = 0.06, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2212, 2209} \[ \frac {2 f^{a+\frac {b}{x}}}{b^2 x \log ^2(f)}-\frac {2 f^{a+\frac {b}{x}}}{b^3 \log ^3(f)}-\frac {f^{a+\frac {b}{x}}}{b x^2 \log (f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int \frac {f^{a+\frac {b}{x}}}{x^4} \, dx &=-\frac {f^{a+\frac {b}{x}}}{b x^2 \log (f)}-\frac {2 \int \frac {f^{a+\frac {b}{x}}}{x^3} \, dx}{b \log (f)}\\ &=\frac {2 f^{a+\frac {b}{x}}}{b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^2 \log (f)}+\frac {2 \int \frac {f^{a+\frac {b}{x}}}{x^2} \, dx}{b^2 \log ^2(f)}\\ &=-\frac {2 f^{a+\frac {b}{x}}}{b^3 \log ^3(f)}+\frac {2 f^{a+\frac {b}{x}}}{b^2 x \log ^2(f)}-\frac {f^{a+\frac {b}{x}}}{b x^2 \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.67 \[ -\frac {f^{a+\frac {b}{x}} \left (b^2 \log ^2(f)-2 b x \log (f)+2 x^2\right )}{b^3 x^2 \log ^3(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 43, normalized size = 0.70 \[ -\frac {{\left (b^{2} \log \relax (f)^{2} - 2 \, b x \log \relax (f) + 2 \, x^{2}\right )} f^{\frac {a x + b}{x}}}{b^{3} x^{2} \log \relax (f)^{3}} \]
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 \[ \int \frac {f^{a + \frac {b}{x}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 73, normalized size = 1.20 \[ \frac {-\frac {x \,{\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b \ln \relax (f )}+\frac {2 x^{2} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}-\frac {2 x^{3} {\mathrm e}^{\left (a +\frac {b}{x}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}}{x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.26, size = 22, normalized size = 0.36 \[ -\frac {f^{a} \Gamma \left (3, -\frac {b \log \relax (f)}{x}\right )}{b^{3} \log \relax (f)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 45, normalized size = 0.74 \[ -\frac {f^{a+\frac {b}{x}}\,\left (\frac {1}{b\,\ln \relax (f)}+\frac {2\,x^2}{b^3\,{\ln \relax (f)}^3}-\frac {2\,x}{b^2\,{\ln \relax (f)}^2}\right )}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 39, normalized size = 0.64 \[ \frac {f^{a + \frac {b}{x}} \left (- b^{2} \log {\relax (f )}^{2} + 2 b x \log {\relax (f )} - 2 x^{2}\right )}{b^{3} x^{2} \log {\relax (f )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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