Optimal. Leaf size=34 \[ -\frac{f^a \left (-b x^3 \log (f)\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-b x^3 \log (f)\right )}{3 x^2} \]
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Rubi [A] time = 0.0208568, antiderivative size = 34, normalized size of antiderivative = 1., 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 = {2218} \[ -\frac{f^a \left (-b x^3 \log (f)\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-b x^3 \log (f)\right )}{3 x^2} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int \frac{f^{a+b x^3}}{x^3} \, dx &=-\frac{f^a \Gamma \left (-\frac{2}{3},-b x^3 \log (f)\right ) \left (-b x^3 \log (f)\right )^{2/3}}{3 x^2}\\ \end{align*}
Mathematica [A] time = 0.003847, size = 34, normalized size = 1. \[ -\frac{f^a \left (-b x^3 \log (f)\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-b x^3 \log (f)\right )}{3 x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 102, normalized size = 3. \begin{align*} -{\frac{{f}^{a}b}{3} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}} \left ({\frac{bx\pi \,\sqrt{3}}{\Gamma \left ({\frac{2}{3}} \right ) }\sqrt [3]{\ln \left ( f \right ) } \left ( -b \right ) ^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}}-{\frac{3\,{{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{2\,{x}^{2}} \left ( -b \right ) ^{-{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}}-{\frac{3\,bx}{2}\sqrt [3]{\ln \left ( f \right ) }\Gamma \left ({\frac{1}{3}},-b{x}^{3}\ln \left ( f \right ) \right ) \left ( -b \right ) ^{-{\frac{2}{3}}}{\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt [3]{-b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21619, size = 38, normalized size = 1.12 \begin{align*} -\frac{\left (-b x^{3} \log \left (f\right )\right )^{\frac{2}{3}} f^{a} \Gamma \left (-\frac{2}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71017, size = 107, normalized size = 3.15 \begin{align*} \frac{\left (-b \log \left (f\right )\right )^{\frac{2}{3}} f^{a} x^{2} \Gamma \left (\frac{1}{3}, -b x^{3} \log \left (f\right )\right ) - f^{b x^{3} + a}}{2 \, x^{2}} \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 \frac{f^{a + b x^{3}}}{x^{3}}\, 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 \frac{f^{b x^{3} + a}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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