Optimal. Leaf size=34 \[ -\frac{x^4 f^a \text{Gamma}\left (\frac{4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}} \]
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Rubi [A] time = 0.0224622, 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} \[ -\frac{x^4 f^a \text{Gamma}\left (\frac{4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{a+b x^3} x^3 \, dx &=-\frac{f^a x^4 \Gamma \left (\frac{4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.0056656, size = 34, normalized size = 1. \[ -\frac{x^4 f^a \text{Gamma}\left (\frac{4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 109, normalized size = 3.2 \begin{align*} -{\frac{{f}^{a}}{3\,b} \left ( -{\frac{2\,x\pi \,\sqrt{3}}{9\,b\Gamma \left ( 2/3 \right ) } \left ( -b \right ) ^{{\frac{4}{3}}}\sqrt [3]{\ln \left ( f \right ) }{\frac{1}{\sqrt [3]{-b{x}^{3}\ln \left ( f \right ) }}}}+{\frac{x{{\rm e}^{b{x}^{3}\ln \left ( f \right ) }}}{b} \left ( -b \right ) ^{{\frac{4}{3}}}\sqrt [3]{\ln \left ( f \right ) }}+{\frac{x}{3\,b} \left ( -b \right ) ^{{\frac{4}{3}}}\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 ) \left ( \ln \left ( f \right ) \right ) ^{-{\frac{4}{3}}}{\frac{1}{\sqrt [3]{-b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.27188, size = 38, normalized size = 1.12 \begin{align*} -\frac{f^{a} x^{4} \Gamma \left (\frac{4}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, \left (-b x^{3} \log \left (f\right )\right )^{\frac{4}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82996, size = 134, normalized size = 3.94 \begin{align*} \frac{3 \, b f^{b x^{3} + a} x \log \left (f\right ) - \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 )}{9 \, b^{2} \log \left (f\right )^{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 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 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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