Optimal. Leaf size=34 \[ -\frac {f^a \Gamma \left (-\frac {1}{3},-b x^3 \log (f)\right ) \sqrt [3]{-b x^3 \log (f)}}{3 x} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2250}
\begin {gather*} -\frac {f^a \sqrt [3]{-b x^3 \log (f)} \text {Gamma}\left (-\frac {1}{3},-b x^3 \log (f)\right )}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rubi steps
\begin {align*} \int \frac {f^{a+b x^3}}{x^2} \, dx &=-\frac {f^a \Gamma \left (-\frac {1}{3},-b x^3 \log (f)\right ) \sqrt [3]{-b x^3 \log (f)}}{3 x}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 34, normalized size = 1.00 \begin {gather*} -\frac {f^a \Gamma \left (-\frac {1}{3},-b x^3 \log (f)\right ) \sqrt [3]{-b x^3 \log (f)}}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs.
\(2(28)=56\).
time = 0.02, size = 100, normalized size = 2.94
method | result | size |
meijerg | \(\frac {f^{a} \left (-b \right )^{\frac {1}{3}} \ln \left (f \right )^{\frac {1}{3}} \left (\frac {3 x^{2} \ln \left (f \right )^{\frac {2}{3}} b \Gamma \left (\frac {2}{3}\right )}{\left (-b \right )^{\frac {1}{3}} \left (-b \,x^{3} \ln \left (f \right )\right )^{\frac {2}{3}}}-\frac {3 \,{\mathrm e}^{b \,x^{3} \ln \left (f \right )}}{x \left (-b \right )^{\frac {1}{3}} \ln \left (f \right )^{\frac {1}{3}}}-\frac {3 x^{2} \ln \left (f \right )^{\frac {2}{3}} b \Gamma \left (\frac {2}{3}, -b \,x^{3} \ln \left (f \right )\right )}{\left (-b \right )^{\frac {1}{3}} \left (-b \,x^{3} \ln \left (f \right )\right )^{\frac {2}{3}}}\right )}{3}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.06, size = 28, normalized size = 0.82 \begin {gather*} -\frac {\left (-b x^{3} \log \left (f\right )\right )^{\frac {1}{3}} f^{a} \Gamma \left (-\frac {1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.10, size = 38, normalized size = 1.12 \begin {gather*} \frac {\left (-b \log \left (f\right )\right )^{\frac {1}{3}} f^{a} x \Gamma \left (\frac {2}{3}, -b x^{3} \log \left (f\right )\right ) - f^{b x^{3} + a}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{a + b x^{3}}}{x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.47, size = 63, normalized size = 1.85 \begin {gather*} \frac {f^a\,\Gamma \left (\frac {2}{3},-b\,x^3\,\ln \left (f\right )\right )\,{\left (-b\,x^3\,\ln \left (f\right )\right )}^{1/3}}{x}-\frac {f^a\,\Gamma \left (\frac {2}{3}\right )\,{\left (-b\,x^3\,\ln \left (f\right )\right )}^{1/3}}{x}-\frac {f^a\,f^{b\,x^3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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