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 = 0.0481365, antiderivative size = 35, normalized size of antiderivative = 1., 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 2214
Rule 2210
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*}
Mathematica [A] time = 0.0052936, 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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Maple [B] time = 0.034, size = 97, normalized size = 2.8 \begin{align*}{\frac{{f}^{a}b\ln \left ( f \right ) }{3} \left ({\frac{{x}^{3}}{b\ln \left ( f \right ) }}+1+3\,\ln \left ( x \right ) -\ln \left ( -b \right ) -\ln \left ( \ln \left ( f \right ) \right ) -{\frac{{x}^{3}}{2\,b\ln \left ( f \right ) } \left ( 2+2\,{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) }+{\frac{{x}^{3}}{b\ln \left ( f \right ) }{{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}+\ln \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) +{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17347, size = 24, normalized size = 0.69 \begin{align*} -\frac{1}{3} \, b f^{a} \Gamma \left (-1, -\frac{b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77106, size = 88, normalized size = 2.51 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{3}}} x^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{3}}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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