Optimal. Leaf size=35 \[ \frac {1}{3} f^{a+\frac {b}{x^3}} x^3-\frac {1}{3} b f^a \text {Ei}\left (\frac {b \log (f)}{x^3}\right ) \log (f) \]
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Rubi [A]
time = 0.03, antiderivative size = 35, 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 = {2245, 2241}
\begin {gather*} \frac {1}{3} x^3 f^{a+\frac {b}{x^3}}-\frac {1}{3} b f^a \log (f) \text {Ei}\left (\frac {b \log (f)}{x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2241
Rule 2245
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^3}} x^2 \, dx &=\frac {1}{3} f^{a+\frac {b}{x^3}} x^3+(b \log (f)) \int \frac {f^{a+\frac {b}{x^3}}}{x} \, dx\\ &=\frac {1}{3} f^{a+\frac {b}{x^3}} x^3-\frac {1}{3} b f^a \text {Ei}\left (\frac {b \log (f)}{x^3}\right ) \log (f)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 32, normalized size = 0.91 \begin {gather*} \frac {1}{3} f^a \left (f^{\frac {b}{x^3}} x^3-b \text {Ei}\left (\frac {b \log (f)}{x^3}\right ) \log (f)\right ) \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. \(96\) vs.
\(2(31)=62\).
time = 0.03, size = 97, normalized size = 2.77
method | result | size |
meijerg | \(\frac {f^{a} b \ln \left (f \right ) \left (-\frac {x^{3} \left (2+\frac {2 b \ln \left (f \right )}{x^{3}}\right )}{2 b \ln \left (f \right )}+\frac {x^{3} {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{b \ln \left (f \right )}+\ln \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )+\expIntegral \left (1, -\frac {b \ln \left (f \right )}{x^{3}}\right )+1+3 \ln \left (x \right )-\ln \left (-b \right )-\ln \left (\ln \left (f \right )\right )+\frac {x^{3}}{\ln \left (f \right ) b}\right )}{3}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 18, normalized size = 0.51 \begin {gather*} -\frac {1}{3} \, b f^{a} \Gamma \left (-1, -\frac {b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 35, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, f^{\frac {a x^{3} + b}{x^{3}}} x^{3} - \frac {1}{3} \, b f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right ) \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 f^{a + \frac {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 = 3.67, size = 33, normalized size = 0.94 \begin {gather*} \frac {f^a\,f^{\frac {b}{x^3}}\,x^3}{3}+\frac {b\,f^a\,\ln \left (f\right )\,\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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