Integrand size = 15, antiderivative size = 207 \[ \int f^{c (a+b x)^n} x^3 \, dx=-\frac {(a+b x)^4 \Gamma \left (\frac {4}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-4/n}}{b^4 n}+\frac {3 a (a+b x)^3 \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-3/n}}{b^4 n}-\frac {3 a^2 (a+b x)^2 \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-2/n}}{b^4 n}+\frac {a^3 (a+b x) \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-1/n}}{b^4 n} \]
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Time = 0.09 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2258, 2239, 2250} \[ \int f^{c (a+b x)^n} x^3 \, dx=\frac {a^3 (a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right )}{b^4 n}-\frac {3 a^2 (a+b x)^2 \left (-c \log (f) (a+b x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right )}{b^4 n}-\frac {(a+b x)^4 \left (-c \log (f) (a+b x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},-c (a+b x)^n \log (f)\right )}{b^4 n}+\frac {3 a (a+b x)^3 \left (-c \log (f) (a+b x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (f)\right )}{b^4 n} \]
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Rule 2239
Rule 2250
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^3 f^{c (a+b x)^n}}{b^3}+\frac {3 a^2 f^{c (a+b x)^n} (a+b x)}{b^3}-\frac {3 a f^{c (a+b x)^n} (a+b x)^2}{b^3}+\frac {f^{c (a+b x)^n} (a+b x)^3}{b^3}\right ) \, dx \\ & = \frac {\int f^{c (a+b x)^n} (a+b x)^3 \, dx}{b^3}-\frac {(3 a) \int f^{c (a+b x)^n} (a+b x)^2 \, dx}{b^3}+\frac {\left (3 a^2\right ) \int f^{c (a+b x)^n} (a+b x) \, dx}{b^3}-\frac {a^3 \int f^{c (a+b x)^n} \, dx}{b^3} \\ & = -\frac {(a+b x)^4 \Gamma \left (\frac {4}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-4/n}}{b^4 n}+\frac {3 a (a+b x)^3 \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-3/n}}{b^4 n}-\frac {3 a^2 (a+b x)^2 \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-2/n}}{b^4 n}+\frac {a^3 (a+b x) \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-1/n}}{b^4 n} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.88 \[ \int f^{c (a+b x)^n} x^3 \, dx=-\frac {(a+b x) \left (-c (a+b x)^n \log (f)\right )^{-4/n} \left ((a+b x)^3 \Gamma \left (\frac {4}{n},-c (a+b x)^n \log (f)\right )-a \left (-c (a+b x)^n \log (f)\right )^{\frac {1}{n}} \left (3 (a+b x)^2 \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (f)\right )+a \left (-c (a+b x)^n \log (f)\right )^{\frac {1}{n}} \left (-3 (a+b x) \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right )+a \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{\frac {1}{n}}\right )\right )\right )}{b^4 n} \]
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\[\int f^{c \left (b x +a \right )^{n}} x^{3}d x\]
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\[ \int f^{c (a+b x)^n} x^3 \, dx=\int { f^{{\left (b x + a\right )}^{n} c} x^{3} \,d x } \]
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\[ \int f^{c (a+b x)^n} x^3 \, dx=\int f^{c \left (a + b x\right )^{n}} x^{3}\, dx \]
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\[ \int f^{c (a+b x)^n} x^3 \, dx=\int { f^{{\left (b x + a\right )}^{n} c} x^{3} \,d x } \]
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\[ \int f^{c (a+b x)^n} x^3 \, dx=\int { f^{{\left (b x + a\right )}^{n} c} x^{3} \,d x } \]
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Timed out. \[ \int f^{c (a+b x)^n} x^3 \, dx=\int f^{c\,{\left (a+b\,x\right )}^n}\,x^3 \,d x \]
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