Optimal. Leaf size=32 \[ -\frac{x f^a \text{Gamma}\left (\frac{1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \]
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Rubi [A] time = 0.0038035, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2208} \[ -\frac{x f^a \text{Gamma}\left (\frac{1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \]
Antiderivative was successfully verified.
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Rule 2208
Rubi steps
\begin{align*} \int f^{a+b x^3} \, dx &=-\frac{f^a x \Gamma \left (\frac{1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}}\\ \end{align*}
Mathematica [A] time = 0.0043517, size = 32, normalized size = 1. \[ -\frac{x f^a \text{Gamma}\left (\frac{1}{3},-b x^3 \log (f)\right )}{3 \sqrt [3]{-b x^3 \log (f)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 78, normalized size = 2.4 \begin{align*}{\frac{{f}^{a}}{3} \left ({\frac{2\,x\pi \,\sqrt{3}}{3\,\Gamma \left ( 2/3 \right ) }\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }{\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}}-{x\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }\Gamma \left ({\frac{1}{3}},-b{x}^{3}\ln \left ( f \right ) \right ){\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt [3]{-b}}}{\frac{1}{\sqrt [3]{\ln \left ( f \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28956, size = 35, normalized size = 1.09 \begin{align*} -\frac{f^{a} x \Gamma \left (\frac{1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, \left (-b x^{3} \log \left (f\right )\right )^{\frac{1}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78871, size = 86, normalized size = 2.69 \begin{align*} \frac{\left (-b \log \left (f\right )\right )^{\frac{2}{3}} f^{a} \Gamma \left (\frac{1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, b \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 + b 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 f^{b x^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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