Optimal. Leaf size=44 \[ -\frac {(a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b \sqrt [3]{-c (a+b x)^3 \log (f)}} \]
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Rubi [A]
time = 0.00, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 = {2239}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
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*}
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Mathematica [A]
time = 0.06, size = 44, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int f^{c \left (b x +a \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.09, size = 60, normalized size = 1.36 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{c \left (a + b x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int f^{c\,{\left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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