Optimal. Leaf size=83 \[ \frac {2 f^{a+\frac {b}{x^3}}}{b^4 \log ^4(f)}-\frac {2 f^{a+\frac {b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)} \]
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Rubi [A] time = 0.09, antiderivative size = 83, 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} \[ \frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {2 f^{a+\frac {b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac {2 f^{a+\frac {b}{x^3}}}{b^4 \log ^4(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2212
Rubi steps
\begin {align*} \int \frac {f^{a+\frac {b}{x^3}}}{x^{13}} \, dx &=-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)}-\frac {3 \int \frac {f^{a+\frac {b}{x^3}}}{x^{10}} \, dx}{b \log (f)}\\ &=\frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)}+\frac {6 \int \frac {f^{a+\frac {b}{x^3}}}{x^7} \, dx}{b^2 \log ^2(f)}\\ &=-\frac {2 f^{a+\frac {b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)}-\frac {6 \int \frac {f^{a+\frac {b}{x^3}}}{x^4} \, dx}{b^3 \log ^3(f)}\\ &=\frac {2 f^{a+\frac {b}{x^3}}}{b^4 \log ^4(f)}-\frac {2 f^{a+\frac {b}{x^3}}}{b^3 x^3 \log ^3(f)}+\frac {f^{a+\frac {b}{x^3}}}{b^2 x^6 \log ^2(f)}-\frac {f^{a+\frac {b}{x^3}}}{3 b x^9 \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 58, normalized size = 0.70 \[ \frac {f^{a+\frac {b}{x^3}} \left (-b^3 \log ^3(f)+3 b^2 x^3 \log ^2(f)-6 b x^6 \log (f)+6 x^9\right )}{3 b^4 x^9 \log ^4(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 60, normalized size = 0.72 \[ \frac {{\left (6 \, x^{9} - 6 \, b x^{6} \log \relax (f) + 3 \, b^{2} x^{3} \log \relax (f)^{2} - b^{3} \log \relax (f)^{3}\right )} f^{\frac {a x^{3} + b}{x^{3}}}}{3 \, b^{4} x^{9} \log \relax (f)^{4}} \]
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 \[ \int \frac {f^{a + \frac {b}{x^{3}}}}{x^{13}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 97, normalized size = 1.17 \[ \frac {\frac {2 x^{12} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{b^{4} \ln \relax (f )^{4}}-\frac {2 x^{9} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{b^{3} \ln \relax (f )^{3}}+\frac {x^{6} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{b^{2} \ln \relax (f )^{2}}-\frac {x^{3} {\mathrm e}^{\left (a +\frac {b}{x^{3}}\right ) \ln \relax (f )}}{3 b \ln \relax (f )}}{x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.28, size = 22, normalized size = 0.27 \[ \frac {f^{a} \Gamma \left (4, -\frac {b \log \relax (f)}{x^{3}}\right )}{3 \, b^{4} \log \relax (f)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.54, size = 60, normalized size = 0.72 \[ -\frac {f^{a+\frac {b}{x^3}}\,\left (\frac {1}{3\,b\,\ln \relax (f)}-\frac {x^3}{b^2\,{\ln \relax (f)}^2}+\frac {2\,x^6}{b^3\,{\ln \relax (f)}^3}-\frac {2\,x^9}{b^4\,{\ln \relax (f)}^4}\right )}{x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 58, normalized size = 0.70 \[ \frac {f^{a + \frac {b}{x^{3}}} \left (- b^{3} \log {\relax (f )}^{3} + 3 b^{2} x^{3} \log {\relax (f )}^{2} - 6 b x^{6} \log {\relax (f )} + 6 x^{9}\right )}{3 b^{4} x^{9} \log {\relax (f )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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