Optimal. Leaf size=92 \[ \frac{a (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b^2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac{(a+b x)^2 \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )}{3 b^2 \left (-c \log (f) (a+b x)^3\right )^{2/3}} \]
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Rubi [A] time = 0.0494645, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2226, 2208, 2218} \[ \frac{a (a+b x) \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )}{3 b^2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac{(a+b x)^2 \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )}{3 b^2 \left (-c \log (f) (a+b x)^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 2226
Rule 2208
Rule 2218
Rubi steps
\begin{align*} \int f^{c (a+b x)^3} x \, dx &=\int \left (-\frac{a f^{c (a+b x)^3}}{b}+\frac{f^{c (a+b x)^3} (a+b x)}{b}\right ) \, dx\\ &=\frac{\int f^{c (a+b x)^3} (a+b x) \, dx}{b}-\frac{a \int f^{c (a+b x)^3} \, dx}{b}\\ &=-\frac{(a+b x)^2 \Gamma \left (\frac{2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^2 \left (-c (a+b x)^3 \log (f)\right )^{2/3}}+\frac{a (a+b x) \Gamma \left (\frac{1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^2 \sqrt [3]{-c (a+b x)^3 \log (f)}}\\ \end{align*}
Mathematica [A] time = 0.0471472, size = 86, normalized size = 0.93 \[ -\frac{(a+b x) \left ((a+b x) \text{Gamma}\left (\frac{2}{3},-c \log (f) (a+b x)^3\right )-a \sqrt [3]{-c \log (f) (a+b x)^3} \text{Gamma}\left (\frac{1}{3},-c \log (f) (a+b x)^3\right )\right )}{3 b^2 \left (-c \log (f) (a+b x)^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{f}^{c \left ( bx+a \right ) ^{3}}x\, 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} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56721, size = 288, normalized size = 3.13 \begin{align*} -\frac{\left (-b^{3} c \log \left (f\right )\right )^{\frac{2}{3}} a \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 ) - \left (-b^{3} c \log \left (f\right )\right )^{\frac{1}{3}} b \Gamma \left (\frac{2}{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^{4} 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}} 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^{{\left (b x + a\right )}^{3} c} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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