Integrand size = 13, antiderivative size = 92 \[ \int f^{c (a+b x)^3} x \, dx=-\frac {(a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^2 \left (-c (a+b x)^3 \log (f)\right )^{2/3}}+\frac {a (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^2 \sqrt [3]{-c (a+b x)^3 \log (f)}} \]
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Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2258, 2239, 2250} \[ \int f^{c (a+b x)^3} x \, dx=\frac {a (a+b x) \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right )}{3 b^2 \sqrt [3]{-c \log (f) (a+b x)^3}}-\frac {(a+b x)^2 \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )}{3 b^2 \left (-c \log (f) (a+b x)^3\right )^{2/3}} \]
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Rule 2239
Rule 2250
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93 \[ \int f^{c (a+b x)^3} x \, dx=-\frac {(a+b x) \left ((a+b x) \Gamma \left (\frac {2}{3},-c (a+b x)^3 \log (f)\right )-a \Gamma \left (\frac {1}{3},-c (a+b x)^3 \log (f)\right ) \sqrt [3]{-c (a+b x)^3 \log (f)}\right )}{3 b^2 \left (-c (a+b x)^3 \log (f)\right )^{2/3}} \]
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\[\int f^{c \left (b x +a \right )^{3}} x d x\]
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none
Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24 \[ \int f^{c (a+b x)^3} x \, dx=-\frac {\left (-b^{3} c \log \left (f\right )\right )^{\frac {2}{3}} a \Gamma \left (\frac {1}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right ) - \left (-b^{3} c \log \left (f\right )\right )^{\frac {1}{3}} b \Gamma \left (\frac {2}{3}, -{\left (b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c\right )} \log \left (f\right )\right )}{3 \, b^{4} c \log \left (f\right )} \]
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\[ \int f^{c (a+b x)^3} x \, dx=\int f^{c \left (a + b x\right )^{3}} x\, dx \]
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\[ \int f^{c (a+b x)^3} x \, dx=\int { f^{{\left (b x + a\right )}^{3} c} x \,d x } \]
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\[ \int f^{c (a+b x)^3} x \, dx=\int { f^{{\left (b x + a\right )}^{3} c} x \,d x } \]
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Timed out. \[ \int f^{c (a+b x)^3} x \, dx=\int f^{c\,{\left (a+b\,x\right )}^3}\,x \,d x \]
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