Optimal. Leaf size=92 \[ \frac {a (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac {(a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^2 \left (-c \log (f) (a+b x)^3\right )^{2/3}} \]
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Rubi [A] time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 2208
Rule 2218
Rule 2226
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*}
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Mathematica [A] time = 0.05, size = 86, normalized size = 0.93 \[ -\frac {(a+b x) \left ((a+b x) \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )-a \sqrt [3]{-c \log (f) (a+b x)^3} \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )\right )}{3 b^2 \left (-c \log (f) (a+b x)^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 114, normalized size = 1.24 \[ -\frac {\left (-b^{3} c \log \relax (f)\right )^{\frac {2}{3}} a \Gamma \left (\frac {1}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \relax (f)\right ) - \left (-b^{3} c \log \relax (f)\right )^{\frac {1}{3}} b \Gamma \left (\frac {2}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \relax (f)\right )}{3 \, b^{4} c \log \relax (f)} \]
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 \[ \int f^{{\left (b x + a\right )}^{3} c} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int x \,f^{\left (b x +a \right )^{3} c}\, dx \]
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 \[ \int f^{{\left (b x + a\right )}^{3} c} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int f^{c\,{\left (a+b\,x\right )}^3}\,x \,d x \]
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 \[ \int f^{c \left (a + b x\right )^{3}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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