Optimal. Leaf size=44 \[ -\frac{(a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b \sqrt [3]{-c \log (f) (a+b x)^3}} \]
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Rubi [A] time = 0.0055393, antiderivative size = 44, 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 = {2208} \[ -\frac{(a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b \sqrt [3]{-c \log (f) (a+b x)^3}} \]
Antiderivative was successfully verified.
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Rule 2208
Rubi steps
\begin{align*} \int f^{c (a+b x)^3} \, dx &=-\frac{(a+b x) \Gamma \left (\frac{1}{3},-c (a+b x)^3 \log (f)\right )}{3 b \sqrt [3]{-c (a+b x)^3 \log (f)}}\\ \end{align*}
Mathematica [A] time = 0.0090916, size = 44, normalized size = 1. \[ -\frac{(a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b \sqrt [3]{-c \log (f) (a+b x)^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{f}^{c \left ( bx+a \right ) ^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{{\left (b x + a\right )}^{3} c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52329, size = 151, normalized size = 3.43 \begin{align*} \frac{\left (-b^{3} c \log \left (f\right )\right )^{\frac{2}{3}} \Gamma \left (\frac{1}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right )}{3 \, b^{3} c \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^{c \left (a + b x\right )^{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^{{\left (b x + a\right )}^{3} c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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