Optimal. Leaf size=35 \[ \frac {1}{3} x^3 f^{a+\frac {b}{x^3}}-\frac {1}{3} b f^a \log (f) \text {Ei}\left (\frac {b \log (f)}{x^3}\right ) \]
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Rubi [A] time = 0.05, 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 = {2214, 2210} \[ \frac {1}{3} x^3 f^{a+\frac {b}{x^3}}-\frac {1}{3} b f^a \log (f) \text {Ei}\left (\frac {b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
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 \[ \frac {1}{3} f^a \left (x^3 f^{\frac {b}{x^3}}-b \log (f) \text {Ei}\left (\frac {b \log (f)}{x^3}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 35, normalized size = 1.00 \[ \frac {1}{3} \, f^{\frac {a x^{3} + b}{x^{3}}} x^{3} - \frac {1}{3} \, b f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x^{3}}\right ) \log \relax (f) \]
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 \[ \int f^{a + \frac {b}{x^{3}}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 97, normalized size = 2.77 \[ \frac {\left (\frac {x^{3} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}}}{b \ln \relax (f )}-\frac {\left (\frac {2 b \ln \relax (f )}{x^{3}}+2\right ) x^{3}}{2 b \ln \relax (f )}+\frac {x^{3}}{b \ln \relax (f )}+\Ei \left (1, -\frac {b \ln \relax (f )}{x^{3}}\right )+3 \ln \relax (x )-\ln \left (-b \right )+\ln \left (-\frac {b \ln \relax (f )}{x^{3}}\right )-\ln \left (\ln \relax (f )\right )+1\right ) b \,f^{a} \ln \relax (f )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 18, normalized size = 0.51 \[ -\frac {1}{3} \, b f^{a} \Gamma \left (-1, -\frac {b \log \relax (f)}{x^{3}}\right ) \log \relax (f) \]
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 \[ \frac {f^a\,f^{\frac {b}{x^3}}\,x^3}{3}+\frac {b\,f^a\,\ln \relax (f)\,\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}{3} \]
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 \[ \int f^{a + \frac {b}{x^{3}}} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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