Optimal. Leaf size=34 \[ -\frac{x^2 f^a \text{Gamma}\left (\frac{2}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{2/3}} \]
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Rubi [A] time = 0.013284, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2218} \[ -\frac{x^2 f^a \text{Gamma}\left (\frac{2}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{a+b x^3} x \, dx &=-\frac{f^a x^2 \Gamma \left (\frac{2}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0049437, size = 34, normalized size = 1. \[ -\frac{x^2 f^a \text{Gamma}\left (\frac{2}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 75, normalized size = 2.2 \begin{align*}{\frac{{f}^{a}}{3} \left ({{x}^{2}\Gamma \left ({\frac{2}{3}} \right ) \left ( -b \right ) ^{{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}} \left ( -b{x}^{3}\ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}}-{{x}^{2} \left ( -b \right ) ^{{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{2}{3}}}\Gamma \left ({\frac{2}{3}},-b{x}^{3}\ln \left ( f \right ) \right ) \left ( -b{x}^{3}\ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}} \right ) \left ( -b \right ) ^{-{\frac{2}{3}}} \left ( \ln \left ( f \right ) \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.29837, size = 38, normalized size = 1.12 \begin{align*} -\frac{f^{a} x^{2} \Gamma \left (\frac{2}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, \left (-b x^{3} \log \left (f\right )\right )^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70051, size = 86, normalized size = 2.53 \begin{align*} \frac{\left (-b \log \left (f\right )\right )^{\frac{1}{3}} f^{a} \Gamma \left (\frac{2}{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}} x\, 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} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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